Recent zbMATH articles in MSC 35Jhttps://zbmath.org/atom/cc/35J2024-09-13T18:40:28.020319ZWerkzeugA mathematical description of the Weber nucleus as a classical and quantum mechanical systemhttps://zbmath.org/1540.340632024-09-13T18:40:28.020319Z"Frauenfelder, Urs"https://zbmath.org/authors/?q=ai:frauenfelder.urs-adrian"Weber, Joa"https://zbmath.org/authors/?q=ai:weber.joaSummary: Wilhelm Weber's electrodynamics is an action-at-a-distance theory which has the property that equal charges inside a critical radius become attractive. Weber's electrodynamics inside the critical radius can be interpreted as a classical Hamiltonian system whose kinetic energy is, however, expressed with respect to a \textit{Lorentzian} metric. In this article we study the Schrödinger equation associated with this Hamiltonian system, and relate it to Weyl's theory of singular Sturm-Liouville problems.Mixed anisotropic and nonlocal Sobolev type inequalities with extremalhttps://zbmath.org/1540.350142024-09-13T18:40:28.020319Z"Garain, Prashanta"https://zbmath.org/authors/?q=ai:garain.prashantaSummary: We establish mixed anisotropic and nonlocal Sobolev type inequalities with an extremal. We show that the extremal function is unique up to a multiplicative constant that is associated with the corresponding mixed anisotropic and nonlocal singular partial differential equation. We prove that such a mixed Sobolev type inequality is necessary and sufficient for the existence of solutions to the associated mixed anisotropic and nonlocal singular partial differential equation. We establish a relation between the purely anisotropic singular equation and mixed anisotropic and nonlocal singular equation.A perturbation result for a Neumann problem in a periodic domainhttps://zbmath.org/1540.350212024-09-13T18:40:28.020319Z"Riva, Matteo Dalla"https://zbmath.org/authors/?q=ai:dalla-riva.matteo"Luzzini, Paolo"https://zbmath.org/authors/?q=ai:luzzini.paolo"Musolino, Paolo"https://zbmath.org/authors/?q=ai:musolino.paoloSummary: We consider a Neumann problem for the Laplace equation in a periodic domain. We prove that the solution depends real analytically on the shape of the domain, on the periodicity parameters, on the Neumann datum, and on its boundary integral.
For the entire collection see [Zbl 1531.35008].Single peak solutions for an elliptic system of FitzHugh-Nagumo typehttps://zbmath.org/1540.350262024-09-13T18:40:28.020319Z"Wang, Bingqi"https://zbmath.org/authors/?q=ai:wang.bingqi"Zhou, Xiangyu"https://zbmath.org/authors/?q=ai:zhou.xiangyu|zhou.xiangyu.1Summary: We study the Dirichlet problem for an elliptic system derived from FitzHugh-Nagummo model as follows:
\[
\begin{cases}
-\varepsilon^2 \Delta u =f(u)-v, & \text{in } \Omega, \\
-\Delta v+\gamma v =\delta_{\varepsilon} u, & \text{in } \Omega, \\
u=v =0, & \text{on } \partial \Omega,
\end{cases}
\]
where \(\Omega\) represents a bounded smooth domain in \(\mathbb{R}^2\) and \(\varepsilon, \gamma\) are positive constants. The parameter \(\delta_{\varepsilon}>0\) is a constant dependent on \(\varepsilon\), and the nonlinear term \(f(u)\) is defined as \(u(u-a)(1-u)\). Here, \(a\) is a function in \(C^2 (\Omega)\cap C^1 (\overline{\Omega})\) with its range confined to \((0,\frac{1}{2})\). Our research focuses on this spatially inhomogeneous scenario whereas the scenario that \(a\) is spatially constant has been studied extensively by many other mathematicians. Specifically, in dimension two, we utilize the Lyapunov-Schmidt reduction method to establish the existence of a single interior peak solution. This is contingent upon a mild condition on \(a\), which acts as an indicator of a location-dependent activation threshold for excitable neurons in the biological environment.Critical sets of solutions of elliptic equations in periodic homogenizationhttps://zbmath.org/1540.350292024-09-13T18:40:28.020319Z"Lin, Fanghua"https://zbmath.org/authors/?q=ai:lin.fang-hua"Shen, Zhongwei"https://zbmath.org/authors/?q=ai:shen.zhongweiSummary: In this paper we study critical sets of solutions \(u_{\varepsilon}\) of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first-order correctors, we show that the \((d-2)\)-dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period \(\varepsilon\), provided that doubling indices for solutions are bounded. The key step is an estimate of ``turning'' of an approximate tangent map, the projection of a non-constant solution \(u_{\varepsilon}\) onto the subspace of spherical harmonics of order \(\ell\), when the doubling index for \(u_{\varepsilon}\) on a sphere \(\partial B(0, r)\) is trapped between \(\ell -\delta\) and \(\ell +\delta\), for \(r\) between 1 and a minimal radius \(r^* \geq C_0 \varepsilon\). This estimate is proved by using harmonic approximation successively. With a suitable \(L^2\) renormalization as well as rescaling we are able to control the accumulated errors introduced by homogenization and projection. Our proof also gives uniform bounds for Minkowski contents of the critical sets.
{\copyright} 2023 Wiley Periodicals LLC.Self-consistent approximations in the theory of composites and their limitationshttps://zbmath.org/1540.350302024-09-13T18:40:28.020319Z"Mityushev, Vladimir"https://zbmath.org/authors/?q=ai:mityushev.vladimir-vSummary: Many attempts were undertaken to modify Maxwell's approach in the theory of composites. Self-consistent methods (effective medium approximation, mean field, Mori-Tanaka methods, reiterated homogenization etc) were advanced to determine the effective properties of composites. It is demonstrated by an example that these extensions are methodologically misleading. They lead to a plenty of illusory different formulas reduced to the Maxwell type, lower order estimation for dilute composites.
For the entire collection see [Zbl 1531.35008].Steady-state bifurcation and spatial patterns of a chemical reaction systemhttps://zbmath.org/1540.350452024-09-13T18:40:28.020319Z"Wang, Jingjing"https://zbmath.org/authors/?q=ai:wang.jingjing"Jia, Yunfeng"https://zbmath.org/authors/?q=ai:jia.yunfeng(no abstract)\(L^\infty\) a-priori estimates for subcritical \(p\)-Laplacian equations with a Carathéodory non-linearityhttps://zbmath.org/1540.350762024-09-13T18:40:28.020319Z"Pardo, Rosa"https://zbmath.org/authors/?q=ai:pardo.rosa-mSummary: Let us consider a quasi-linear boundary value problem \(-\Delta_p u = f(x, u)\), in \(\Omega\), with Dirichlet boundary conditions, where \(\Omega\subset\mathbb{R}^N\), with \(p < N\), is a bounded smooth domain strictly convex, and the non-linearity \(f\) is a Carathéodory function \(p\)-super-linear and subcritical. We provide \(L^\infty\) a priori estimates for weak solutions, in terms of their \(L^{p^\ast}\)-norm, where \(p^\ast = \frac{Np}{N-p}\) is the critical Sobolev exponent. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in elliptic regularity for the \(p\)-Laplacian combined either with Gagliardo-Nirenberg or with Caffarelli-Kohn-Nirenberg interpolation inequalities. By a subcritical non-linearity we mean, for instance, \(|f(x,s)| \leq |x|^{-\mu}\tilde{f}(s),\) where \(\mu\in(0, p)\), and \(\tilde{f}(s)/|s|^{p_\mu^\ast - 1}\rightarrow 0\) as \(|s|\rightarrow\infty\), here \(p^\ast_\mu := \frac{p(N - \mu)}{N-p}\) is the critical Hardy-Sobolev exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when \(f(x, s) = |x|^{-\mu}\frac{|s|^{p^\ast_\mu - 2}s}{[\log(e + |s|)]^\alpha}\), with \(\mu\in[1, p)\), then, for any \(\varepsilon > 0\) there exists a constant \(C_\varepsilon > 0\) such that for any solution \(u\in H^1_0(\Omega)\), the following holds
\[
[\log(e+\|u\|_\infty)]^\alpha \leq C_\varepsilon(1 + \|u\|_{p^\ast})^{(p^\ast_\mu - p)(1+\varepsilon)},
\]
where \(C_\varepsilon\) is independent of the solution \(u\).Sharp Strichartz type estimates for the Schrödinger equation associated with harmonic oscillatorhttps://zbmath.org/1540.350772024-09-13T18:40:28.020319Z"Senapati, P. Jitendra Kumar"https://zbmath.org/authors/?q=ai:senapati.p-jitendra-kumar"Boggarapu, Pradeep"https://zbmath.org/authors/?q=ai:boggarapu.pradeepIn this paper, by utilizing purely the \(L^2\to L^p\) operator norm estimates of the spectral projections associated to the harmonic oscillator proved in [\textit{E. Jeong} et al., ``Endpoint eigenfunction bounds for the Hermite operator'', Preprint, \url{arXiv:2205.03036}] and [\textit{H. Koch} and \textit{D. Tataru}, Duke Math. J. 128, No. 2, 369--392 (2005; Zbl 1075.35020)], the authors obtain the Strichartz-type inequality for the solution to the Schrödinger equation associated with the harmonic oscillator. Furthermore, the Strichartz-type estimates are sharp in sense of regularity of initial data.
Reviewer: Jiqiang Zheng (Beijing)Maximum principles for elliptic operators in unbounded Riemannian domainshttps://zbmath.org/1540.350792024-09-13T18:40:28.020319Z"Bisterzo, Andrea"https://zbmath.org/authors/?q=ai:bisterzo.andreaSummary: The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.Liouville-type theorems for a nonlinear fractional Choquard equationhttps://zbmath.org/1540.350812024-09-13T18:40:28.020319Z"Duong, Anh Tuan"https://zbmath.org/authors/?q=ai:duong.anh-tuan"Loan, Tran Thi"https://zbmath.org/authors/?q=ai:tran-thi-loan."Quyet, Dao Trong"https://zbmath.org/authors/?q=ai:quyet.dao-trong"Thang, Dao Manh"https://zbmath.org/authors/?q=ai:thang.dao-manhSummary: In this paper, we are concerned with the fractional Choquard equation on the whole space \(\mathbb{R}^N\)
\[
(-\Delta )^s u=\left(\frac{1}{|x|^{N-2s}}*u^p \right) u^{p-1}
\]
with \(0<s<1, N>2s\) and \(p\in \mathbb{R}\). We first prove that the equation does not possess any positive solution for \(p\leq 1\). When \(p>1\), we establish a Liouville type theorem saying that if
\[
N<6s+\frac{4s(1+\sqrt{p^2 -p})}{p-1},
\]
then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation.
{{\copyright} 2023 Wiley-VCH GmbH.}Liouville-type theorem for logarithmic Laplacian systems in coercive epigraphshttps://zbmath.org/1540.350822024-09-13T18:40:28.020319Z"Liu, Baiyu"https://zbmath.org/authors/?q=ai:liu.baiyu"Xu, Shasha"https://zbmath.org/authors/?q=ai:xu.shashaSummary: This paper is concerned with the Dirichlet problem of logarithmic Laplacian equations and systems on unbounded regions. We first establish new narrow region principles for the logarithmic Laplacian equation and system in coercive epigraphs separately. Then by using the direct method of moving planes, we obtain a monotonicity result for the solutions of logarithmic Laplacian equations and a Liouville-type theorem for solutions of logarithmic Laplacian systems in coercive epigraphs.The Kotake-Narasimhan theorem in general ultradifferentiable classeshttps://zbmath.org/1540.350842024-09-13T18:40:28.020319Z"Fürdös, Stefan"https://zbmath.org/authors/?q=ai:furdos.stefanSummary: We prove a Kotake-Narasimhan type theorem in general ultradifferentiable classes given by weight matrices. In doing so we simultaneously recover and generalize significantly the known results for classes given by weight sequences and weight functions. In particular, we obtain a sharp Kotake-Narasimhan theorem for Beurling classes.Lipschitz continuity of the solutions to the Dirichlet problems for the invariant Laplacianshttps://zbmath.org/1540.350862024-09-13T18:40:28.020319Z"Liu, Congwen"https://zbmath.org/authors/?q=ai:liu.congwen"Xu, Heng"https://zbmath.org/authors/?q=ai:xu.hengSummary: This short note is motivated by an attempt to understand the distinction between the Laplace operator and the hyperbolic Laplacian on the unit ball of \(\mathbb{R}^n\), regarding the Lipschitz continuity of the solutions to the corresponding Dirichlet problems. We investigate the Dirichlet problem
\[\begin{cases}
\Delta_\vartheta u = 0, \ & \text{in } \mathbb{B}^n, \\
u = \phi, & \text{on } \mathbb{S}^{n - 1},
\end{cases}\]
where
\[
\Delta_\vartheta : = (1 - | x |^2) \bigg \{\frac{1 - | x |^2}{4} \Delta + \vartheta \sum_{j = 1}^n x_j \frac{\partial}{\partial x_j} + \vartheta \left(\frac{n}{2} - 1 - \vartheta \right) I \bigg\}.
\]
We show that: (i) when \(\vartheta > 0\), the Lipschitz continuity of boundary data always implies the Lipschitz continuity of the solutions; (ii) when \(\vartheta = 0\), there exists a Lipschitz continuous function \(\phi : \mathbb{S}^{n - 1} \to \mathbb{R}\) such that the solution is not Lipschitz continuous; (iii) when \(\vartheta < 0\), the solution is definitely not Lipschitz continuous unless \(\phi \equiv 0\).Improved quadrature formulas for the direct value of the normal derivative of a single-layer potentialhttps://zbmath.org/1540.351072024-09-13T18:40:28.020319Z"Krutitskii, P. A."https://zbmath.org/authors/?q=ai:krutitskii.pavel-a"Reznichenko, I. O."https://zbmath.org/authors/?q=ai:reznichenko.igor-oSummary: A single-layer potential for the Helmholtz equation in the three-dimensional case and a single-layer potential for the Laplace equation are considered. A quadrature rule is derived for the direct value of the normal derivative of the single-layer potential with a continuous density given on a closed or open surface. The quadrature rule provides a much higher accuracy than previously available formulas, which is confirmed by numerical tests. The quadrature rule can be used for the numerical solution of boundary value problems for Laplace and Helmholtz equations by applying the boundary integral equation method.On a quadrature formula for the direct value of the double layer potentialhttps://zbmath.org/1540.351082024-09-13T18:40:28.020319Z"Reznichenko, Igor O."https://zbmath.org/authors/?q=ai:reznichenko.igor-o"Krutitskii, Pavel A."https://zbmath.org/authors/?q=ai:krutitskii.pavel-a"Kolybasova, Valentina V."https://zbmath.org/authors/?q=ai:kolybasova.v-vSummary: A quadrature formula for the direct value of the double layer potential with continuous density given on a closed or open surface is derived. The double layer potential for the Helmholtz equations are considered, the potential for the Laplace equation is a particular case. The proposed quadrature formula gives significantly higher accuracy than standard quadrature formula, which is confirmed by numerical tests. The derived quadrature formula can be used for numerical solving boundary value problems for the Laplace and Helmholtz equations by the method of potentials and boundary integral equations.
For the entire collection see [Zbl 1531.35008].Integral representations for second-order elliptic systems in the planehttps://zbmath.org/1540.351092024-09-13T18:40:28.020319Z"Soldatov, A. P."https://zbmath.org/authors/?q=ai:soldatov.aleksandr-pavlovichSummary: A fundamental solution matrix for elliptic systems of the second order with constant leading coefficients is constructed. It is used to obtain an integral representation of functions belonging to the Hölder class in a closed domain with a Lyapunov boundary. In the case of an infinite domain, these functions have power-law asymptotics at infinity. The representation is used to study a mixed-contact boundary value problem for a second-order elliptic system with piecewise constant leading coefficients. The problem is reduced to a system of integral equations that are Fredholm in the domain and singular at its boundary.On \(\infty\)-ground states in the planehttps://zbmath.org/1540.351132024-09-13T18:40:28.020319Z"Lindgren, Erik"https://zbmath.org/authors/?q=ai:lindgren.erik"Lindqvist, Peter"https://zbmath.org/authors/?q=ai:lindqvist.peterSummary: We study \(\infty\)-Ground states in convex domains in the plane. In a polygon, the points where an \(\infty\)-Ground state does not satisfy the \(\infty\)-Laplace Equation are characterized: they are restricted to lie on specific curves, which are acting as attracting (fictitious) streamlines. The gradient is continuous outside these curves and no streamlines can meet there.Analysis of the transmission eigenvalue problem with two conductivity parametershttps://zbmath.org/1540.351202024-09-13T18:40:28.020319Z"Ceja Ayala, Rafael"https://zbmath.org/authors/?q=ai:ayala.rafael-ceja.1"Harris, Isaac"https://zbmath.org/authors/?q=ai:harris.isaac"Kleefeld, Andreas"https://zbmath.org/authors/?q=ai:kleefeld.andreas"Pallikarakis, Nikolaos"https://zbmath.org/authors/?q=ai:pallikarakis.nikolaos(no abstract)Localization for general Helmholtzhttps://zbmath.org/1540.351212024-09-13T18:40:28.020319Z"Cheng, Xinyu"https://zbmath.org/authors/?q=ai:cheng.xinyu"Li, Dong"https://zbmath.org/authors/?q=ai:li.dong"Yang, Wen"https://zbmath.org/authors/?q=ai:yang.wenSummary: In [Commun. Contemp. Math. 25, No. 2, Article ID 2250016, 18 p. (2023; Zbl 1509.35344)], \textit{V. Guan} et al. established the equivalence of the classical Helmholtz equation with a ``fractional Helmholtz'' equation in which the Laplacian operator is replaced by the nonlocal fractional Laplacian operator. More general equivalence results are obtained for symbols which are complete Bernstein and satisfy additional regularity conditions. In this work we introduce a novel and general set-up for this Helmholtz equivalence problem. We show that under very mild and easy-to-check conditions on the Fourier multiplier, the general Helmholtz equation can be effectively reduced to a localization statement on the support of the symbol.Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layerhttps://zbmath.org/1540.351222024-09-13T18:40:28.020319Z"Chesnel, Lucas"https://zbmath.org/authors/?q=ai:chesnel.lucas"Nazarov, Sergei A."https://zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovich"Taskinen, Jari"https://zbmath.org/authors/?q=ai:taskinen.jariSummary: We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector \(\kappa =(\kappa_1, \kappa_2)\) which characterises the domain at the edges. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases \(\kappa_1 \geq 0\) and \(\kappa_2 \in [-\kappa_1, \kappa_1]\). We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to \(\kappa\). In particular, we show that for a given \(\kappa_1 >0\), there is some \(h(\kappa_1) >0\) such that discrete spectrum exists for \(\kappa_2 \in [-\kappa_1, 0) \cup (h(\kappa_1), \kappa_1]\) whereas it is empty for \(\kappa_2 \in [0, h(\kappa_1)]\). The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.Hausdorff measure bounds for nodal sets of Steklov eigenfunctionshttps://zbmath.org/1540.351232024-09-13T18:40:28.020319Z"Decio, Stefano"https://zbmath.org/authors/?q=ai:decio.stefanoSummary: We study nodal sets of Steklov eigenfunctions in a bounded domain with \(\mathcal{C}^2\) boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that, for \(u_{\lambda}\) a Steklov eigenfunction with eigenvalue \(\lambda\neq 0\), we have \(\mathcal{H}^{d-1}(\{u_{\lambda}=0\})\geq c_{\Omega}\), where \(c_{\Omega}\) is independent of \(\lambda\). We also prove an almost sharp upper bound, namely, \(\mathcal{H}^{d-1}(\{u_{\lambda}=0\})\leq C_{\Omega}\lambda\log (\lambda +e)\).On explicit numerically realizable formulae for Poincaré-Steklov operatorshttps://zbmath.org/1540.351242024-09-13T18:40:28.020319Z"Demidov, Aleksander S."https://zbmath.org/authors/?q=ai:demidov.aleksander-s(no abstract)Nonstandard boundary value problems of theory of two-dimensional vector fieldshttps://zbmath.org/1540.351252024-09-13T18:40:28.020319Z"Dubinskii, Yu. A."https://zbmath.org/authors/?q=ai:dubinskii.yulii-a"Provorotova, L. V."https://zbmath.org/authors/?q=ai:provorotova.l-vSummary: We study nonstandard boundary value problems of field theory for the system of Poisson equations. The feature of these problems is that the solution is looked for in subspaces that are kernels of trace operators or functionals. We establish the existence and uniqueness of a weak solution and classify the nonstandard boundary value problems under consideration.An interface formulation for the Poisson equation in the presence of a semiconducting single-layer materialhttps://zbmath.org/1540.351262024-09-13T18:40:28.020319Z"Jourdana, Clément"https://zbmath.org/authors/?q=ai:jourdana.clement"Pietra, Paola"https://zbmath.org/authors/?q=ai:pietra.paolaSummary: In this paper, we consider a semiconducting device with an active zone made of a single-layer material. The associated Poisson equation for the electrostatic potential (to be solved in order to perform self-consistent computations) is characterized by a surface particle density and an out-of-plane dielectric permittivity in the region surrounding the single-layer. To avoid mesh refinements in such a region, we propose an interface problem based on the natural domain decomposition suggested by the physical device. Two different interface continuity conditions are discussed. Then, we write the corresponding variational formulations adapting the so called three-fields formulation for domain decomposition and we approximate them using a proper finite element method. Finally, numerical experiments are performed to illustrate some specific features of this interface approach.Multi-phase \(k\)-quadrature domains and applications to acoustic waves and magnetic fieldshttps://zbmath.org/1540.351272024-09-13T18:40:28.020319Z"Kow, Pu-Zhao"https://zbmath.org/authors/?q=ai:kow.pu-zhao"Shahgholian, Henrik"https://zbmath.org/authors/?q=ai:shahgholian.henrikSummary: The primary objective of this paper is to explore the multi-phase variant of quadrature domains associated with the Helmholtz equation, commonly referred to as \(k\)-quadrature domains. Our investigation employs both the minimization problem approach, which delves into the segregation ground state of an energy functional, and the partial balayage procedure, drawing inspiration from the recent work by Gardiner and Sjödin. Furthermore, we present practical applications of these concepts in the realms of acoustic waves and magnetic fields.A coupled marching method for Cauchy problems of the Helmholtz equation in complex waveguideshttps://zbmath.org/1540.351282024-09-13T18:40:28.020319Z"Li, Peng"https://zbmath.org/authors/?q=ai:li.peng.25"Liu, Keying"https://zbmath.org/authors/?q=ai:liu.keyingSummary: This paper develops a coupled marching method to solve Cauchy problems of the Helmholtz equation in waveguides with irregularly varying local parts. We reformulate the spectral projection marching method (SPMM) to act as a coupler to couple itself with the operator marching method (OMM). In marching computations, the coupled marching method applies the SPMM to compute wave propagations in irregular areas, and applies the OMM to slowly varying parts. We also present transformation error estimates for analyzing errors arising in the coordinate transformation between different local subspaces, and discuss principles for applying the coupled marching method. Extensive numerical experiments demonstrate the accuracy, convergence and efficiency of the coupled marching method in various complex waveguides.Stability of grating diffraction problems for plane wave incidence: explicit dependence on wavenumbers and incident angleshttps://zbmath.org/1540.351292024-09-13T18:40:28.020319Z"Zhu, Linlin"https://zbmath.org/authors/?q=ai:zhu.linlin"Hu, Guanghui"https://zbmath.org/authors/?q=ai:hu.guanghuiSummary: Suppose a time-harmonic plane wave is incident onto an impenetrable grating profile of Dirichlet or Impedance kind in two dimensions. The grating profile is assumed to be given by a Lipschitz function. We derive a stability estimate of this grating diffraction problem using the variational method with a transparent boundary condition. An explicit dependence of solutions on the incident wavenumber and the incident angle is obtained, which carries over to transmission problems for penetrable gratings. Our approach relies heavily on using Rellich's identities in periodic structures.The Morse property of limit functions appearing in mean field equations on surfaces with boundaryhttps://zbmath.org/1540.351302024-09-13T18:40:28.020319Z"Hu, Zhengni"https://zbmath.org/authors/?q=ai:hu.zhengni"Bartsch, Thomas"https://zbmath.org/authors/?q=ai:bartsch.thomas.1|bartsch.thomas.2Let \(\Sigma\) denote a smooth and compact surface with boundary \(\partial \Sigma\) of a Riemannian manifold with metric \(g\). Denote also by \(G^g\) the Green function of the Laplace-Beltrami operator with Neuman boundary condition.
This article discusses Morse properties of functions of the form
\[
f_g(x)=\sum_{1\leq i\leq m} \sigma_i^2 R^g(x_i)+\sum_{1\leq i, j\leq m, i\neq j}\sigma_i \sigma_j G^g(x_i, x_j)+h(x_1, x_2, \dots, x_m),
\]
where \(R^g\) is the corresponding Robin function of \(G^g\) and \(h\) is any \(C^2\) arbitrary function. The main result of the article establishes that for any Riemannian metric \(g\), there exists a metric \(\tilde g\) arbitrarily close to \(g\) and in the conformal class of with the property that \(f_{\tilde g}\) is a is a Morse function. As a byproduct, it is obtained that if all \(\sigma_i>0\), then the set of all Riemannian metrics \(g\) for which \(f_g\) is a Morse function becomes an open and dense set in the set of all Riemannian metrics.
Reviewer: Marius Ghergu (Dublin)On the non-degeneracy of the Robin function for the fractional Laplacian on symmetric domainshttps://zbmath.org/1540.351312024-09-13T18:40:28.020319Z"Ortega, Alejandro"https://zbmath.org/authors/?q=ai:ortega.alejandroSummary: In this work we prove, under symmetry and convexity assumptions on the domain \(\Omega\), the non-degeneracy at zero of the Hessian matrix of the Robin function for the spectral fractional Laplacian. This work extends to the fractional setting the results of \textit{M. Grossi} [C. R., Math., Acad. Sci. Paris 335, No. 2, 157--160 (2002; Zbl 1011.58017)] concerning the classical Laplace operator.The best constant of the \(L^p\) Sobolev-type inequality corresponding to elliptic operator in \(\mathbf{R}^N\)https://zbmath.org/1540.351322024-09-13T18:40:28.020319Z"Yamagishi, Hiroyuki"https://zbmath.org/authors/?q=ai:yamagishi.hiroyuki"Kametaka, Yoshinori"https://zbmath.org/authors/?q=ai:kametaka.yoshinoriSummary: The \(L^p\) Sobolev-type inequality shows that the supremum of \(|u(y)|\) defined on \(\mathbb{R}^N\) is estimated from above by constant \(C\) multiples of the \(L^p\) norm of \((-\varDelta+a^2)u(x)\). Among such constant \(C\), the smallest constant is the best constant \(C_0\). If we replace \(C\) by \(C_0\) in the \(L^p\) Sobolev-type inequality, then the equality holds for the best function \(U(x)\). The aim of this paper is to find \(C_0\) and \(U(x)\) of the \(L^p\) Sobolev-type inequality. The Green function \(G(x-y)\) of partial differential equation of elliptic type \((-\varDelta +a^2)u(x)=f(x)\) defined on \(\mathbb{R}^N\) is an important factor in this paper because \(C_0\) and \(U(x)\) consist of the Green function.Existence of positive solutions for weakly coupled Schrödinger system with supercritical growthhttps://zbmath.org/1540.351332024-09-13T18:40:28.020319Z"An, Xiaoming"https://zbmath.org/authors/?q=ai:an.xiaoming"Deng, Yinbin"https://zbmath.org/authors/?q=ai:deng.yinbin"Yang, Xian"https://zbmath.org/authors/?q=ai:yang.xianSummary: This paper focuses on the existence of positive solutions for the following weakly coupled Schrödinger system with supercritical growth except at the origin:
\[
\begin{cases}
-\Delta u_1 = \mu_1|u_1|^{p(r) - 2}u_1 + \beta|u_2|^{\frac{p(r)}{2}}|u_1|^{\frac{p(r)}{2}-2}u_1, \quad & x\in B_1(0), \\
-\Delta u_2 = \mu_2|u_2|^{p(r) - 2}u_2 + \beta|u_1|^{\frac{p(r)}{2}}|u_2|^{\frac{p(r)}{2}-2}u_2, \quad & x\in B_1(0),
\end{cases}
\]
where \(B_1(0)\) is an unit ball \({\mathbb{R}^N}\) with \(N\ge 3\), \(\beta\in\mathbb{R}\) is a coupling constant, \( \mu_1,\mu_2\in\mathbb{R}\) are constants, \( p(r) = 2^* + r^{\alpha}\) with \(2^* = \frac{2N}{N-2} \). Assuming that \(0<\alpha<\min\{\frac{N}{2},N-2\} \), we apply concentration-compactness idea to show that the problem has a positive solution provided that \(\beta>0\) if \(N\ge 5\) and \(\beta\in(0,\beta_0]\cup[\beta_1,+\infty)\) for some positive constants \(\beta_0<\beta_1\) if \(N = 3,4 \).Magnetic Schrödinger operator with the potential supported in a curved two-dimensional striphttps://zbmath.org/1540.351342024-09-13T18:40:28.020319Z"Bory-Reyes, Juan"https://zbmath.org/authors/?q=ai:bory-reyes.juan"Barseghyan, Diana"https://zbmath.org/authors/?q=ai:barseghyan.diana"Schneider, Baruch"https://zbmath.org/authors/?q=ai:schneider.baruchSummary: We consider the magnetic Schrödinger operator \(H = (i\nabla +A)^2 - V\) with a non-negative potential \(V\) supported over a strip which is a local deformation of a straight one, and the magnetic field \(B := \operatorname{rot}(A)\) is assumed to be non-zero and local. We show that the magnetic field does not change the essential spectrum of this system, and investigate a sufficient condition for the discrete spectrum of \(H\) to be empty.Generalized Schrödinger operators on the Heisenberg group and Hardy spaceshttps://zbmath.org/1540.351352024-09-13T18:40:28.020319Z"Bui, The Anh"https://zbmath.org/authors/?q=ai:the-anh-bui."Hong, Qing"https://zbmath.org/authors/?q=ai:hong.qing"Hu, Guorong"https://zbmath.org/authors/?q=ai:hu.guorongSummary: Let \(L = - \Delta_{\mathbb{H}^n} + \mu\) be a generalized Schrödinger operator on the Heisenberg group \(\mathbb{H}^n\), where \(\Delta_{\mathbb{H}^n}\) is the sub-Laplacian, and \(\mu\) is a nonnegative Radon measure satisfying certain conditions. In this paper, we first establish some estimates of the fundamental solution and the heat kernel of \(L\). Applying these estimates, we then study the Hardy spaces \(H_L^1( \mathbb{H}^n)\) introduced in terms of the maximal function associated with the heat semigroup \(e^{- t L} \); in particular, we obtain an atomic decomposition of \(H_L^1( \mathbb{H}^n)\), and prove the Riesz transform characterization of \(H_L^1( \mathbb{H}^n)\). The dual space of \(H_L^1( \mathbb{H}^n)\) is also studied.Sharp global well-posedness for the cubic nonlinear Schrödinger equation with third order dispersionhttps://zbmath.org/1540.351362024-09-13T18:40:28.020319Z"Carvajal, X."https://zbmath.org/authors/?q=ai:carvajal.xavier-paredes"Panthee, M."https://zbmath.org/authors/?q=ai:panthee.mahendraSummary: We consider the initial value problem (IVP) associated to the cubic nonlinear Schrödinger equation with third-order dispersion
\[
\partial_t u + i\alpha\partial^2_xu - \partial^3_xu + i\beta|u|^2u = 0, \quad x,t\in\mathbb{R},
\]
for given data in the Sobolev space \(H^s(\mathbb{R})\). This IVP is known to be locally well-posed for given data with Sobolev regularity \(s > -\frac{1}{4}\) and globally well-posed for \(s \geq 0\) [the first author, Electron. J. Differ. Equ. 2004, Paper No. 13, 10 p. (2004; Zbl 1051.35084)].
For given data in \(H^s(\mathbb{R})\), \(0 > s > -\frac{1}{4}\) no global well-posedness result is known. In this work, we derive an \textit{almost conserved quantity} for such data and obtain a sharp global well-posedness result. Our result answers the question left open in [loc. cit.].Two-sided heat kernel estimates for Schrödinger operators with unbounded potentialshttps://zbmath.org/1540.351372024-09-13T18:40:28.020319Z"Chen, Xin"https://zbmath.org/authors/?q=ai:chen.xin.13|chen.xin.1|chen.xin.4|chen.xin.10|chen.xin.7|chen.xin.2|chen.xin.8|chen.xin.9|chen.xin.3|chen.xin"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.65|wang.jian.1|wang.jian.10|wang.jian.52|wang.jian.70|wang.jian.3|wang.jian.63|wang.jian.16|wang.jian.2|wang.jian.28|wang.jian.18|wang.jian.29|wang.jian.44|wang.jian.9|wang.jian.15|wang.jian.55|wang.jian.71|wang.jian.78|wang.jian.5|wang.jian.13|wang.jian.57|wang.jian.12|wang.jian.68|wang.jian.21|wang.jian.6|wang.jian.4Summary: Consider the Schrödinger operator \(\mathcal{L}^V = - \Delta + V\) on \(\mathbb{R}^d\), where \(V : \mathbb{R}^d \to [0, \infty)\) is a nonnegative and locally bounded potential on \(\mathbb{R}^d\) so that for all \(x \in \mathbb{R}^d\) with \(| x | \geq 1\), \(c_1 g(| x |) \leq V(x) \leq c_2 g(| x |)\) with some constants \(c_1, c_2 > 0\) and a nondecreasing and strictly positive function \(g : [0, \infty) \to [1, + \infty)\) that satisfies \(g(2 r) \leq c_0 g(r)\) for all \(r > 0\) and \(\lim_{r \to \infty} g(r) = \infty \). We establish global in time and qualitatively sharp bounds for the heat kernel of the associated Schrödinger semigroup by the probabilistic method. In particular, we can present global in space and time two-sided bounds of heat kernel even when the Schrödinger semigroup is not intrinsically ultracontractive. Furthermore, two-sided estimates for the corresponding Green's function are also obtained.Existence of a positive bound state solution for logarithmic Schrödinger equationhttps://zbmath.org/1540.351382024-09-13T18:40:28.020319Z"Feng, Weixun"https://zbmath.org/authors/?q=ai:feng.weixun"Tang, Xianhua"https://zbmath.org/authors/?q=ai:tang.xian-hua"Zhang, Luyu"https://zbmath.org/authors/?q=ai:zhang.luyuSummary: We investigate the existence of bound state solutions for the logarithmic Schrödinger equation
\[
-\Delta u+V(x)u = u \log u^2 \text{ in } \mathbb{R}^N, N \geq 1,
\]
where \(V(x) \in C(\mathbb{R}^N)\) has a limit at infinity. In the case when the ground state is not attained, we construct a higher critical value for the variational functional. The classical variational methods cannot be applied directly to deal with the above problem due to nonsmoothness. To recover the smoothness, we use the superlinear power-law perturbation of the logarithmic nonlinearity.On the eigenfunctions of the essential spectrum of the model problem for the Schrödinger operator with singular potentialhttps://zbmath.org/1540.351392024-09-13T18:40:28.020319Z"Lyalinov, Mikhail A."https://zbmath.org/authors/?q=ai:lyalinov.mikhail-anatolievichSummary: We are concerned with generalized eigenfunctions of the continuous (essential) spectrum for the Schrödinger operator with singular \(\delta\)-potential that has support on the sides of an angle in the plane. Operators of this kind appear in quantum-mechanical models for quantum state destruction of two point-interacting quantum particles of which one is reflected by a potential barrier. We propose an approach capable of constructing integral representations for eigenfunctions in terms of the solution of a functional-difference equation with spectral parameter. Solutions of this equation are studied by reduction to an integral equation, with the subsequent study of the spectral properties of the corresponding integral operator. We also construct an asymptotic formula for the eigenfunction at large distances. For this formula a physical interpretation from the point of view of wave scattering is given.
Our approach can be used to deal with eigenfunctions in a broad class of related problems for the Schrödinger operator with singular potential.Resolvent bounds for Lipschitz potentials in dimension two and higher with singularities at the originhttps://zbmath.org/1540.351402024-09-13T18:40:28.020319Z"Obovu, Donnell"https://zbmath.org/authors/?q=ai:obovu.donnellSummary: We consider, for \(h, E>0\), the semiclassical Schrödinger operator \(-h^2 \Delta+V-E\) in dimension two and higher. The potential \(V\) and its radial derivative \(\partial_r V\) are bounded away from the origin, have long-range decay and \(V\) is bounded by \(r^{-\delta}\) near the origin while \(\partial_r V\) is bounded by \(r^{-1-\delta}\), where \(0\leq \delta <4(\sqrt{2}-1)\). In this setting, we show that the resolvent bound is exponential in \(h^{-1}\), while the exterior resolvent bound is linear in \(h^{-1}\).Zero-mass gauged Schrödinger equations with supercritical exponential growthhttps://zbmath.org/1540.351412024-09-13T18:40:28.020319Z"Shen, Liejun"https://zbmath.org/authors/?q=ai:shen.liejunSummary: We study the following gauged nonlinear Schrödinger equation
\[
\begin{cases}
- \Delta u + \left( \int\limits_{| x |}^\infty \frac{h_u ( s )}{s} u^2 ( s ) d s + \frac{h_u^2 ( | x | )}{| x |^2} \right) u = f ( u ) - a | u |^{p - 2} u, \\
u ( x ) = u ( | x | ),
\end{cases}
\]
where \(a > 0\), \(p \in(1, 2)\), \(h_u(s) = \int_0^s \frac{r}{2} u^2(r) d r\) and \(f\) possesses the supercritical exponential growth in the Trudinger-Moser sense at infinity. Via introducing a new type of Trudinger-Moser inequality in a suitable work space here, we shall exploit the general minimax principle and elliptic regular result to investigate the existence of mountain-pass type solutions for the equation using variational method.Multiplicity of normalized solutions for Schrödinger equation with mixed nonlinearityhttps://zbmath.org/1540.351422024-09-13T18:40:28.020319Z"Xu, Lin"https://zbmath.org/authors/?q=ai:xu.lin|xu.lin.3"Song, Changxiu"https://zbmath.org/authors/?q=ai:song.changxiu"Xie, Qilin"https://zbmath.org/authors/?q=ai:xie.qilinSummary: In this paper, we explore the multiplicity of normalized solutions for Schrödinger equation with mixed nonlinearities
\[
\begin{cases}
-\Delta u+V(\epsilon x)u = \lambda u+\mu |u|^{q-2} u+|u|^{p-2} u & \text{in }\mathbb{R}^N, \\
\int_{\mathbb{R}^N} |u|^2 \, dx = c,
\end{cases}
\]
where \(\mu >0\), \(c>0\), \(2<q<2+4/N<p<2N/(N-2)\), \(N \geq 3\), \(\epsilon >0\) is a parameter and \(\lambda \in \mathbb{R}\) is an unknown parameter that appears as a Lagrange multiplier. The potential \(V\) is a bounded and continuous nonnegative function, satisfying some suitable global conditions. By employing the minimization techniques and the truncated argument, we obtain that the number of normalized solutions is not less than the number of global minimum points of \(V\) when the parameter \(\epsilon\) is sufficiently small.Normalized solutions for nonlinear Schrödinger equations on graphshttps://zbmath.org/1540.351432024-09-13T18:40:28.020319Z"Yang, Yunyan"https://zbmath.org/authors/?q=ai:yang.yunyan"Zhao, Liang"https://zbmath.org/authors/?q=ai:zhao.liang.10Summary: We are concerned with the nonlinear Schrödinger equation with an \(L^2\) mass constraint on both finite and locally finite graphs and prove that the equation has a normalized solution by employing variational methods. We also pay attention to the behaviours of the normalized solution as the mass constraint tends to \(0^+\) or \(+\infty\) and give clear descriptions of the limit equations. Finally, we provide some numerical experiments on a finite graph to illustrate our theoretical results.Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentialshttps://zbmath.org/1540.351442024-09-13T18:40:28.020319Z"Zhang, Caifeng"https://zbmath.org/authors/?q=ai:zhang.caifeng"Zhu, Maochun"https://zbmath.org/authors/?q=ai:zhu.maochunSummary: The purpose of this paper is three-fold. First, we establish singular Trudinger-Moser inequalities with less restrictive constraint:
\[
\underset{u\in H^1 (\mathbb{R}^2),\int\limits_{\mathbb{R}^2}(\vert\nabla u\vert^2 +V(x)u^2)\mathrm{d}x\leq 1}{\mathrm{sup}} \int\limits_{\mathbb{R}^2}\frac{e^{4\pi \left(1-\frac{\beta}{2}\right)u^2}-1}{\vert x\vert^{\beta}} \mathrm{d}x<+\infty,
\tag{0.1}
\]
where \(0<\beta <2\), \(V(x)\geq 0\) and may vanish on an open set in \(\mathbb{R}^2\). Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in \(\mathbb{R}^2\):
\[
-\Delta u+\gamma u=\frac{f(u)}{\vert x\vert^{\beta}},
\tag{0.2}
\]
where the nonlinearity \(f\) has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in [\textit{C. Zhang} and \textit{L. Chen}, Adv. Nonlinear Stud. 18, No. 3, 567--585 (2018; Zbl 1397.35117)]
and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in \(\mathbb{R}^2\):
\[
-\Delta u+V(x)u=\frac{f(u)}{\vert x\vert^{\beta}}.
\tag{0.3}
\]Space-time mixed norm estimates in Riemannian symmetric spaces of non-compact typehttps://zbmath.org/1540.351452024-09-13T18:40:28.020319Z"Zhang, Hong-Wei"https://zbmath.org/authors/?q=ai:zhang.hongwei.1|zhang.hongwei|zhang.hongwei.2Summary: We present a summary of recent advances in the Strichartz inequality and the smoothing property on non-compact type and general rank Riemannian symmetric spaces.
For the entire collection see [Zbl 1537.35003].Local behavior, radial symmetry and classification of solutions to weighted elliptic equationshttps://zbmath.org/1540.351462024-09-13T18:40:28.020319Z"Li, Kui"https://zbmath.org/authors/?q=ai:li.kui"Zhang, Zhitao"https://zbmath.org/authors/?q=ai:zhang.zhitaoSummary: We study positive solutions with an isolated singularity to a class of weighted elliptic equations in \(B_1\backslash\{0\}\) and in \(\mathbb{R}^N\backslash\{0\} \). First, in \(B_1\backslash\{0\}\) we present new results on the asymptotic behavior at the singular point for positive solutions. Then in \(\mathbb{R}^N\backslash\{0\} \), we prove radially symmetric properties for positive singular solutions, and give a complete classification for these solutions.Asymptotics of harmonic functions in the absence of monotonicity formulashttps://zbmath.org/1540.351472024-09-13T18:40:28.020319Z"Li, Zongyuan"https://zbmath.org/authors/?q=ai:li.zongyuanSummary: In this article, we study the asymptotics of harmonic functions. In literature, one typical method is by proving the monotonicity of a version of rescaled Dirichlet energies, and use it to study the renormalized solution -- the Almgren's blowup. However, such monotonicity formulas require strong smoothness assumptions on domains and operators. We are interested in the cases when monotonicity formulas are not available, including variable coefficient equations with unbounded lower order terms, Dirichlet problems on rough (non-\(C^1)\) domains, and Robin problems with rough Robin potentials.
For the entire collection see [Zbl 1537.35003].Normalized solutions to the quasilinear Schrödinger equations with combined nonlinearitieshttps://zbmath.org/1540.351482024-09-13T18:40:28.020319Z"Mao, Anmin"https://zbmath.org/authors/?q=ai:mao.anmin"Lu, Shuyao"https://zbmath.org/authors/?q=ai:lu.shuyaoSummary: We consider the radially symmetric positive solutions to quasilinear problem
\[
-\Delta u - u \Delta u^2 + \lambda u=f(u), \quad \text{in } \mathbb{R}^N,
\]
having prescribed mass \(\int_{\mathbb{R}^N} |u|^2 = a^2\), where \(a > 0\) is a constant, \(\lambda\) appears as a Lagrange multiplier. We focus on the pure \(L^2\)-supercritical case and combination case of \(L^2\)-subcritical and \(L^2\)-supercritical nonlinearities
\[
f(u) = \tau |u|^{q-2} u+|u|^{p-2} u, \quad \tau > 0, \qquad \text{where } 2 < q < 2+\frac{4}{N} \quad \text{and} \quad p > \bar{p},
\]
where \(\bar{p} := 4+\frac{4}{N}\) is the \(L^2\)-critical exponent. Our work extends and develops some recent results in the literature.Weighted Choquard equation perturbed with weighted nonlocal termhttps://zbmath.org/1540.351492024-09-13T18:40:28.020319Z"Singh, Gurpreet"https://zbmath.org/authors/?q=ai:singh.gurpreetSummary: We investigate the following problem
\[
-\mathrm{div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \left( |x|^{-\theta}*\frac{|u|^b}{|x|^{\alpha}}\right) \frac{|u|^{b-2}}{|x|^{\alpha}} u+\lambda \left( |x|^{-\gamma}*\frac{|u|^c}{|x|^{\beta}}\right) \frac{|u|^{c-2}}{|x|^{\beta}}u \quad \text{in }\mathbb{R}^N,
\]
where \(b, c, \alpha, \beta >0\), \(\theta,\gamma \in (0,N)\), \(N\geq 3\), \(2\leq m< \infty\) and \(\lambda \in\mathbb{R}\). Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.Extremal functions for a singular super-critical Trudinger-Moser inequalityhttps://zbmath.org/1540.351502024-09-13T18:40:28.020319Z"Zhao, Juan"https://zbmath.org/authors/?q=ai:zhao.juanSummary: In this paper, we deal with a singular super-critical Trudinger-Moser inequality on a unit ball of \(\mathbb{R}^n\), \(n \geq 3\). For any \(p > 1\), we set
\[
\lambda_p(\mathbb{B}) = \inf_{u \in W^{1, n}_0(\mathbb{B}), u\not\equiv0}\frac{\int_{\mathbb{B}}|\nabla u|^n dx}{(\int_{\mathbb{B}}|u|^pdx)^{n/p}}
\]
as an eigenvalue related to the \(n\)-Laplacian. Let \(\mathscr{S}\) be a set of radially symmetric functions. Precisely, if \(\beta\geq 0\) and \(\alpha < (1 + \frac{p}{n}\beta)^{n -1+n/p}\lambda_p(\mathbb{B})\), then there exists a positive constant \(\epsilon_0\) such that when \(\lambda \leq 1 + \varepsilon 0\),
\[
\sup_{u\in W^{1, n}_0(\mathbb{B}) \cap\mathscr{S}, \int_{\mathbb{B}}|\nabla u|^ndx - \alpha(\int_{\mathbb{B}}|u|^p|x|^{p\beta}dx)^{\frac{n}{p}} \leq1} \int_{\mathbb{B}} |x|^{p\beta}\left(e^{\alpha_n (1+ \frac{p}{n}\beta)|u|^{\frac{n}{n-1}}} - \lambda\sum^m_{k=0}\frac{|\alpha_n(1 + \frac{p}{n}\beta)u^{\frac{n}{n-1}}|^k}{k!}\right) dx
\]
is attained, where \(\alpha_n = n\omega^{1/(n-1)}_{n-1}\), \(\omega_{n-1}\) is the surface area of the unit ball in \(\mathbb{R}^n\). The proof is based on the method of blow-up analysis. The case \(\lambda = 0\) was studied by \textit{Y. Yang} and \textit{X. Zhu} in [Arch. Math. 112, No. 5, 531--545 (2019; Zbl 1426.35105)]. \textit{ D. G. de Figueiredo} [Proc. Am. Math. Soc. 144, No. 8, 3369--3380 (2016; Zbl 1341.35080)] considered the case \(p = 2\), \(\beta \geq 0\), and \(\alpha = 0\) in two dimension. The case \(\lambda = 0\), \(p = n\), \(-1 < \beta < 0\), and \(\alpha = 0\) was considered by \textit{S. K. Adimurthi} [NoDEA, Nonlinear Differ. Equ. Appl. 13, No. 5--6, 585--603 (2007; Zbl 1171.35367)]. Our results extend those of the above cases.The Poincaré inequality for 3D-vector fields and the Neumann problemhttps://zbmath.org/1540.351512024-09-13T18:40:28.020319Z"Dubinskii, Yu. A."https://zbmath.org/authors/?q=ai:dubinskii.yulii-a"Zubkov, P. V."https://zbmath.org/authors/?q=ai:zubkov.p-vSummary: For 3D-vector fields we obtain a family of integral inequalities that can be regarded as the Poincaré inequality within the framework of field theory. We establish a connection between solutions to the corresponding integral identities and the solution to the Neumann problem.On the functional \(\int_\Omega f + \int_{\Omega^\ast} G\)https://zbmath.org/1540.351522024-09-13T18:40:28.020319Z"Guang, Qiang"https://zbmath.org/authors/?q=ai:guang.qiang"Li, Qi-Rui"https://zbmath.org/authors/?q=ai:li.qirui"Wang, Xu-Jia"https://zbmath.org/authors/?q=ai:wang.xu-jiaSummary: In this paper, we consider a class of functionals subject to a duality restriction. The functional is of the form \(\mathcal{J} (\Omega, \Omega^\ast)=\int_{\Omega} f + \int_{\Omega^\ast} g\), where \(f, g\) are given nonnegative functions in a manifold. The duality is a relation \(\alpha (x, y) \leq 0 \; \forall \;x \in \Omega, y \in \Omega^\ast\), for a suitable function \(\alpha\). This model covers several geometric and physical applications. In this paper we review two topological methods introduced in the study of the functional, and discuss possible extensions of the methods to related problems.Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergencehttps://zbmath.org/1540.351532024-09-13T18:40:28.020319Z"Boccardo, Lucio"https://zbmath.org/authors/?q=ai:boccardo.lucio"Ildefonso Díaz, Jesús"https://zbmath.org/authors/?q=ai:diaz-diaz.jesus-ildefonso"Gómez-Castro, David"https://zbmath.org/authors/?q=ai:gomez-castro.davidSummary: In this article we study the existence, uniqueness, and integrability of solutions to the Dirichlet problem \(-\mathrm{div}(M(x) \nabla u) = -\mathrm{div} (E(x) u) + f\) in a bounded domain of \(\mathbb{R}^N\) with \(N \geq 3\). We are particularly interested in singular \(E\) with \(\mathrm{div}\, E \geq 0\). We start by recalling known existence results when \(|E| \in L^N\) that do not rely on the sign of \(\mathrm{div}\, E\). Then, under the assumption that \(\mathrm{div}\, E \geq 0\) distributionally, we extend the existence theory to \(|E| \in L^2\). For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of \(E\) singular at one point as \(Ax /|x|^2\), or towards the boundary as \(\mathrm{div}\, E \sim \mathrm{dist}(x, \partial \Omega)^{-2-\alpha}\). In these cases the singularity of \(E\) leads to \(u\) vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. \(\partial u/\partial n < 0\), fails in the presence of such singular drift terms \(E\).Laplace-Beltrami equation on Lipschitz hypersurfaces in the generic Bessel potential spaceshttps://zbmath.org/1540.351542024-09-13T18:40:28.020319Z"Duduchava, Roland"https://zbmath.org/authors/?q=ai:duduchava.rolandSummary: The purpose of the present short note is to expose a new approach to the investigation of boundary value problems (BVPs) for the Laplace-Beltrami equation on a hypersurface \(\mathcal{S}\subset \mathbb{R}^3\) with Lipschitz boundary \(\Gamma =\partial \mathcal{S}\), containing a finite number of angular points (nodes) \(c_j\) of magnitude \(\alpha_j, j=1,2,\ldots ,n\). The Dirichlet, Neumann and mixed type BVPs are considered in a non-classical setting when solutions are sought in the generic Bessel potential spaces (GBPS) \(\mathbb{G}\mathbb{H}^s_p(\mathcal{S},\rho)\), \(s>1/p\), \(1<p<\infty\) with weight \(\rho (t)=\prod \limits_{j=1}^n|t-c_j|{}^{\gamma_j}\) (the definition see below). By a localization procedure, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed BVPs for the Laplace equation in a planar angular domain \(\Omega_{\alpha_j}\subset \mathbb{R}^2\) of magnitude \(\alpha_j\), \(j=1,2\ldots ,n\). Further the model problem in the GBPS with weight \(\mathbb{G}\mathbb{H}^s_p(\Omega_{\alpha_j},t^{\gamma_j})\) is investigated by means of Mellin convolution operators on the semi-axes \(\mathbb{R}^+=(0,\infty)\). Explicit criteria for the Fredholm property and the unique solvability of the initial BVPs are obtained and singularities of solutions at nodes to the mentioned BVPs are indicated. In contrast to the results on the same BVPs in the classical Bessel potential spaces \(\mathbb{H}^s_p(\mathcal{S})\), the Fredholm property in the GBPS \(\mathbb{G}\mathbb{H}^s_p(\mathcal{S},\rho)\) with weight is independent of the smoothness parameter \(s\) and Fredholm conditions as well as singularities of solutions are indicated very explicitly.
For the entire collection see [Zbl 1537.35003].Boundary integral equation methods for Lipschitz domains in linear elasticityhttps://zbmath.org/1540.351552024-09-13T18:40:28.020319Z"Le Louër, Frédérique"https://zbmath.org/authors/?q=ai:le-louer.frederiqueSummary: A review of stable boundary integral equation methods for solving the Navier equation with either Dirichlet or Neumann boundary conditions in the exterior of a Lipschitz domain is presented. The conventional combined-field integral equation (CFIE) formulations, that are used to avoid spurious resonances, do not give rise to a coercive variational formulation for nonsmooth geometries anymore. To circumvent this issue, either the single layer or the double layer potential operator is composed with a compact or a Steklov-Poincaré type operator. The later can be constructed from the well-know elliptic boundary integral operators associated to the Laplace equation and Gårding's inequalities are satisfied. Some Neumann interior eigenvalue computations for the unit square and cube are presented for forthcoming numerical investigations.The \(L^p\)-continuity of wave operators for higher order Schrödinger operatorshttps://zbmath.org/1540.351562024-09-13T18:40:28.020319Z"Erdoğan, M. Burak"https://zbmath.org/authors/?q=ai:erdogan.mehmet-burak"Green, William R."https://zbmath.org/authors/?q=ai:green.william-rThe authors study the \(L^p\) boundedness of the wave operators, which are defined by
\[
W_{\pm}=s-\lim_{t\rightarrow \pm\infty}e^{itH}e^{-itH_0},
\]
where \(H_0=(-\Delta)^m\) is the free higher order Schrödinger operator and \(H=(-\Delta)^m+V(x)\) is the perturbed operator. Here, \(m\in\mathbb N\), \(m>1\), and \(V\) is a real-valued, decaying potential. The authors restrict their focus to the case when the spatial dimension \(n >2m\) and they show first that, for a small potential, the wave operators extend to bounded operators on \(L^p(\mathbb R^n)\) for all \(1\leq p\leq \infty\). Further, they show that if the smallness assumption is removed in even dimensions the wave operators remain bounded for \(1 <p <\infty\). Finally, with slightly more decay of the potential, they recover the endpoints \(p =1, \infty\) in odd dimensions, that is, for \(n>2m\) odd, the wave operators extend to bounded operators on \(L^p(\mathbb R^n)\) for all \(1\leq p\leq \infty\).
Reviewer: Said El Manouni (Riyadh)Non-existence results for fourth order Hardy-Hénon equations in dimensions 2 and 3https://zbmath.org/1540.351572024-09-13T18:40:28.020319Z"Tran, Thi Ngoan"https://zbmath.org/authors/?q=ai:tran.thi-ngoan"Ngô, Quốc Anh"https://zbmath.org/authors/?q=ai:ngo-quoc-anh."Tran, Van Tuan"https://zbmath.org/authors/?q=ai:tran.van-tuanSummary: We consider the fourth order Hardy-Hénon equation
\[\Delta^2 u = | x |^\sigma u^p \text{ in } \mathbb{R}^n\]
with \(n \geq 2\), \(\sigma > - 4\), and \(p > 1\). This is the fourth order analogy of the second order equation \(- \Delta u = | x |^\sigma u^p\), known as the Hardy-Hénon equation, which was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above fourth order equation, the assumption \(n \geq 4\) is often assumed. In this work, we are interested in classical solutions to the equation in the case of low dimensions, namely \(2 \leq n \leq 3\). Here by a classical solution \(u\) we mean \(u\) belongs to the class
\[
C( \mathbb{R}^n) \cap C^4( \mathbb{R}^n \setminus \{0 \}) \text{ if } \sigma < 0 \text{ and } C^4( \mathbb{R}^n) \text{ if } \sigma \geq 0.
\]
We show that if \(2 \leq n \leq 3\) then the equation admits no classical solution. In fact, we are able to provide a single treatment for \(2 \leq n \leq 4\).Existence results for some anisotropic elliptic problems having variable exponent and \(L^1\)-datahttps://zbmath.org/1540.351582024-09-13T18:40:28.020319Z"Benboubker, Mohamed Badr"https://zbmath.org/authors/?q=ai:benboubker.mohamed-badr"Hjiaj, Hassane"https://zbmath.org/authors/?q=ai:hjiaj.hassaneSummary: In this paper, we propose to study the existence of entropy solutions for the strongly nonlinear anisotropic elliptic equation
\[Au+ H(x,u,\nabla u) = f\qquad \mbox{in } \Omega,\]
where \(f\) belongs to \(L^1(\Omega)\), \(A\) is a Leray-Lions operator and \(H\) is a nonlinear lower order term with nonstandard growth with respect to \(|\nabla u|\) (i.e., such that \(|H(x,s,\xi)|\leq c(x) + b(|s|)\sum_{i=1}^N|\xi_i|^{p_i(x)}),\) but without assuming the sign condition \(H(x,s,\xi)s\geq0\). A concrete example is given to illustrate the existence result.Existence and multiplicity of nontrivial solutions to a class of elliptic Kirchhoff-Boussinesq type problemshttps://zbmath.org/1540.351592024-09-13T18:40:28.020319Z"Carlos, Romulo D."https://zbmath.org/authors/?q=ai:carlos.romulo-d"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcherSummary: We consider the following class of elliptic Kirchhoff-Boussinesq type problems given by
\[
\Delta^2 u \pm \Delta_p u = f(u) + \beta|u|^{2_{\ast\ast}-2}u \text{ in }\Omega\text{ and }\Delta u = u = 0 \text{ on } \partial\Omega,
\]
where \(\Omega\subset\mathbb{R}^N\) is a bounded and smooth domain, \(2 < p \leq \frac{2N}{N-2}\) for \(N \geq 3\), \(2_{\ast\ast} = \frac{2N}{N-4}\) if \(N \geq 5\), \(2_{\ast\ast} = \infty\) if \(3 \leq N <5\) and \(f\) is a continuous function. We show existence and multiplicity of nontrivial solutions using minimization technique on the Nehari manifold, Mountain Pass Theorem and Genus theory. In this paper we consider the subcritical case \(\beta = 0\) and the critical case \(\beta =1\).A unified weighted inequality for fourth-order partial differential operators and applicationshttps://zbmath.org/1540.351602024-09-13T18:40:28.020319Z"Cui, Yan"https://zbmath.org/authors/?q=ai:cui.yan"Fu, Xiaoyu"https://zbmath.org/authors/?q=ai:fu.xiaoyu"Tian, Jiaxin"https://zbmath.org/authors/?q=ai:tian.jiaxinSummary: In this paper, we establish a fundamental inequality for fourth order partial differential operator \(\mathcal{P} \overset{\triangle}{=} \alpha \partial_s + \beta \partial_{ss} + \Delta^2 (\alpha, \beta \in \mathbb{R})\) with an abstract exponential-type weight function. Such kind of weight functions including not only the regular weight functions but also the singular weight functions. Using this inequality we are able to prove some Carleman estimates for the operator \(\mathcal{P}\) with some suitable boundary conditions in the case of \(\beta <0\) or \(\alpha \neq 0\), \(\beta = 0\). As application, we obtain a resolvent estimate for \(\mathcal{P}\), which can imply a log-type stabilization result for the plate equation with clamped boundary conditions or hinged boundary conditions.Analysis of a Poisson-Nernst-Planck-Fermi system for charge transport in ion channelshttps://zbmath.org/1540.351612024-09-13T18:40:28.020319Z"Jüngel, Ansgar"https://zbmath.org/authors/?q=ai:jungel.ansgar"Massimini, Annamaria"https://zbmath.org/authors/?q=ai:massimini.annamariaSummary: A modified Poisson-Nernst-Planck system in a bounded domain with mixed Dirichlet-Neumann boundary conditions is analyzed. It describes the concentrations of ions immersed in a polar solvent and the correlated electric potential due to the ion-solvent interaction. The concentrations solve cross-diffusion equations, which are thermodynamically consistent. The considered mixture is saturated, meaning that the sum of the ion and solvent concentrations is constant. The correlated electric potential depends nonlocally on the electric potential and solves the fourth-order Poisson-Fermi equation. The existence of global bounded weak solutions is proved by using the boundedness-by-entropy method. The novelty of the paper is the proof of the weak-strong uniqueness property. In contrast to the existence proof, we include the solvent concentration in the cross-diffusion system, leading to a diffusion matrix with nontrivial kernel. Then the proof is based on the relative entropy method for the extended cross-diffusion system and the positive definiteness of a related diffusion matrix on a subspace.Towards the optimality of the ball for the Rayleigh conjecture concerning the clamped platehttps://zbmath.org/1540.351622024-09-13T18:40:28.020319Z"Leylekian, Roméo"https://zbmath.org/authors/?q=ai:leylekian.romeoSummary: The first eigenvalue of the Dirichlet bilaplacian shall be interpreted as the principal frequency of a vibrating plate with clamped boundary. In 1894, Rayleigh conjectured that, upon prescribing the area, the vibrating clamped plate with least principal frequency is circular. In 1995, \textit{N. S. Nadirashvili} [Arch. Ration. Mech. Anal. 129, No. 1, 1--10 (1995; Zbl 0826.73035)] proved the Rayleigh Conjecture. Subsequently, \textit{M. S. Ashbaugh} and \textit{R. D. Benguria} [Duke Math. J. 78, No. 1, 1--17 (1995; Zbl 0833.35035)] proved the analogue of the conjecture in dimension 3. Since then, the conjecture has remained open in dimension \(d>3\). In this document, we contribute in answering the conjecture in high dimension under a particular assumption regarding the critical values of the optimal eigenfunction. More precisely, we prove that if the optimal eigenfunction has no critical value except its minimum and maximum, then the conjecture holds. This is performed thanks to an improvement of Talenti's comparison principle, made possible after a fine study of the geometry of the eigenfunction's nodal domains.Multiplicity results for \(p(x)\)-biharmonic equations with nonlinear boundary conditionshttps://zbmath.org/1540.351632024-09-13T18:40:28.020319Z"Rasouli, S. H."https://zbmath.org/authors/?q=ai:rasouli.sayyed-hashem|rasouli.seyyed-hashem(no abstract)On the triple junction problem without symmetry hypotheseshttps://zbmath.org/1540.351642024-09-13T18:40:28.020319Z"Alikakos, Nicholas D."https://zbmath.org/authors/?q=ai:alikakos.nicholas-d"Geng, Zhiyuan"https://zbmath.org/authors/?q=ai:geng.zhiyuanSummary: We investigate the Allen-Cahn system \(\Delta u-W_u(u)=0, u: \mathbb{R}^2 \to \mathbb{R}^2\), where \(W \in C^2 (\mathbb{R}^2, [0,+\infty))\) is a potential with three global minima. We establish the existence of an entire solution \(u\) which possesses a triple junction structure. The main strategy is to study the global minimizer \(u_\varepsilon\) of the variational problem \(\min \int_{B_1} \left(\frac{\varepsilon}{2} |\nabla u|^2+\frac{1}{\varepsilon} W(u) \right) \,\mathrm{d}z, u=g_\varepsilon\) on \(\partial B_1\) for some suitable boundary data \(g_\varepsilon\). The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution or on the potential.Ground state solutions of a Choquard type system with critical exponential growthhttps://zbmath.org/1540.351652024-09-13T18:40:28.020319Z"Chen, Wenjing"https://zbmath.org/authors/?q=ai:chen.wenjing"Tang, Huan"https://zbmath.org/authors/?q=ai:tang.huan(no abstract)Monotonic convergence of positive radial solutions for general quasilinear elliptic systemshttps://zbmath.org/1540.351662024-09-13T18:40:28.020319Z"Devine, Daniel"https://zbmath.org/authors/?q=ai:devine.daniel"Karageorgis, Paschalis"https://zbmath.org/authors/?q=ai:karageorgis.paschalisSummary: We study the asymptotic behavior of positive radial solutions for quasilinear elliptic systems that have the form
\[
\begin{cases}
\Delta_pu=c_1|x|^{m_1}\cdot g_1(v)\cdot |\nabla u|^\alpha\quad &\text{in }\mathbb{R}^n,\\
\Delta_pv=c_2|x|^{m_2}\cdot g_2(v)\cdot g_3(|\nabla u|)\quad &\text{in }\mathbb{R}^n,
\end{cases}
\]
where \(\Delta_p\) denotes the \(p\)-Laplace operator, \(p>1\), \(n\geqslant 2\), \(c_1,c_2>0\) and \(m_1,m_2,\alpha\geqslant 0\). For a general class of functions \(g_j\) which grow polynomially, we show that every non-constant positive radial solution \((u,v)\) asymptotically approaches \((u_0,v_0)=(C_\lambda|x|^\lambda,c_\mu|x|^\mu)\) for some parameters \(\lambda,\mu,C_\lambda,c_\mu>0\). In fact, the convergence is monotonic in the sense that both \(u/u_0\) and \(v/v_0\) are decreasing. We also obtain similar results for more general systems.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}On the existence of solutions for nonlinear Schrödinger-Poisson systemhttps://zbmath.org/1540.351672024-09-13T18:40:28.020319Z"dos Passos Corrêa, Genivaldo"https://zbmath.org/authors/?q=ai:dos-passos-correa.genivaldo"dos Santos, Gelson C. G."https://zbmath.org/authors/?q=ai:dos-santos.gelson-conceicao-g|dos-santos.gelson-c-g"Silva, Julio Roberto S."https://zbmath.org/authors/?q=ai:silva.julio-roberto-sSummary: In this paper, we study the following class of Schrödinger-Poisson systems
\[
\begin{cases}
-\epsilon^2 \Delta u+V(x) u + \phi u = f(u) \text{ in } \mathbb{R}^3, \\
-\epsilon^2 \Delta\phi = u^2 \text{ in } \mathbb{R}^3, \\
u \in H^1 (\mathbb{R}^3), \phi \in D^{1,2} (\mathbb{R}^3),
\end{cases}
\]
where \(\epsilon >0\) is a real parameter, \(V : \mathbb{R}^3 \longrightarrow \mathbb{R}\) and \(f : \mathbb{R} \to \mathbb{R}\) are continuous functions. The nonlinearity \(f\) has a subcritical growth and \(V\) is a positive potential that satisfies following hypothesis: or \(V\) satisfies ``\textit{a condition of type Palais-Smale}'' or there is a bounded domain \(\Omega\) in \(\mathbb{R}^3\) such that \(V\) has no critical point in \(\partial\Omega\).
The technique used to prove the main result combined \textit{V. Benci} and \textit{D. Fortunato}'s reduction method [Topol. Methods Nonlinear Anal. 11, No. 2, 283--293 (1998; Zbl 0926.35125)] with \textit{M. A. del Pino} and \textit{P. L. Felmer}'s approach [Calc. Var. Partial Differ. Equ. 4, No. 2, 121--137 (1996; Zbl 0844.35032)]) and some ideas from [\textit{C. O. Alves}, J. Elliptic Parabol. Equ. 1, No. 2, 231--241 (2015; Zbl 1378.35142)]) and [\textit{H. Brézis} and \textit{T. Kato}, J. Math. Pures Appl., IX. Sér. 58, 137--151 (1979; Zbl 0408.35025)].Positive solutions for Kirchhoff-type elliptic system with critical exponent in \(\mathbb{R}^3\)https://zbmath.org/1540.351682024-09-13T18:40:28.020319Z"Liu, Tianhao"https://zbmath.org/authors/?q=ai:liu.tianhaoSummary: We consider the following coupled elliptic system with critical exponent:
\[
\begin{cases}
-\left( a_i + b_i \sum\limits_{j=1}^d b_j \int_{\Omega} |\nabla u_j |^2 \right) \Delta u_i = \lambda_i u_i + \mu_i |u_i |^4 u_i + \sum\limits_{j=1, j \neq i}^d \beta_{ij} |u_j |^3 |u_i | u_i, \\
u_i \in H_0^1 (\Omega), i = 1, 2, \ldots, d.
\end{cases}
\]
Here, \(d \geq 2\) is an integer and \(\Omega\subset\mathbb{R}^3\) is a smooth bounded domain, \(a_i, \lambda_i, \mu_i >0, b_i \geq 0\) for all \(i=1, \ldots, d, \beta_{ij} = \beta_{ji} \leq 0\) for \(i, j = 1, \ldots, d, i \neq j\). Note that the nonlinearity and the coupling terms are both critical in dimension three (that is, Sobolev critical exponent \(2^{\ast} = \frac{2N}{N-2}=6\) when \(N=3)\). In this paper, we obtain the existence of positive solutions for this critical system by variational arguments. Besides, we study the concentration behaviors of these positive solutions as \(\mathbf{b} \to 0\) and \(\beta_{ij} \to 0^-\) for \(i \neq j\), where \(\mathbf{b} = (b_1, \ldots, b_d)\) is a vector.Existence and multiplicity of solutions for the Schrödinger-Poisson system with indefinite nonlinearityhttps://zbmath.org/1540.351692024-09-13T18:40:28.020319Z"Pi, Huirong"https://zbmath.org/authors/?q=ai:pi.huirong"Zhu, Xinxin"https://zbmath.org/authors/?q=ai:zhu.xinxinSummary: We consider the existence of solutions for the following Schrödinger-Poisson system with indefinite nonlinearity
\[
\begin{cases}
- \Delta u + u + \mu \phi u = a (x) |u|^{p - 2} u + \lambda k(x) u, \; x \in \mathbb{R}^3, \\
- \Delta \phi = u^2, \; \phi \in D^{1, 2} (\mathbb{R}^3),
\end{cases}
\]
where \(p \in (2, 4)\) and the parameters \(\mu, \lambda > 0\), the functions \(0 < k(x) \in L^{\frac{3}{2}}(\mathbb{R}^3)\) and \(a(x) \in C (\mathbb{R}^3, \mathbb{R})\) satisfying \(a_\infty := \lim\limits_{|x| \to \infty} a(x) < 0 < \max\limits_{x \in \mathbb{R}^3} a(x) =: a_{\max}\) and other suitable conditions. We prove the existence and multiplicity results depending on \(\mu\), \(\lambda\) and \(p\). We also study the asymptotic behavior of solutions when the parameter \(\mu \to 0^+\).Existence of high energy solutions for superlinear coupled Klein-Gordons and Born-Infeld equationshttps://zbmath.org/1540.351702024-09-13T18:40:28.020319Z"Wang, Lixia"https://zbmath.org/authors/?q=ai:wang.lixia"Zhao, Pingping"https://zbmath.org/authors/?q=ai:zhao.pingping"Zhang, Dong"https://zbmath.org/authors/?q=ai:zhang.dongSummary: In this article, we study the system of Klein-Gordon and Born-Infeld equations
\[\begin{gathered} -\Delta u +V(x)u-(2\omega+\phi)\phi u =f(x,u), \quad x\in \mathbb{R}^3,\\
\Delta \phi+\beta\Delta_4\phi=4\pi(\omega+\phi)u^2, \quad x\in \mathbb{R}^3, \end{gathered}\]
where \(\Delta_4\phi=\mathrm{div}(|\nabla\phi|^2\nabla\phi)\), \(\omega\) is a positive constant. Assuming that the primitive of \(f(x,u)\) is of 2-superlinear growth in \(u\) at infinity, we prove the existence of multiple solutions using the fountain theorem. Here the potential \(V\) are allowed to be a sign-changing function.Double phase systems with convex-concave nonlinearity on complete manifoldhttps://zbmath.org/1540.351712024-09-13T18:40:28.020319Z"Aberqi, Ahmed"https://zbmath.org/authors/?q=ai:aberqi.ahmed"Benslimane, Omar"https://zbmath.org/authors/?q=ai:benslimane.omar"Knifda, Mohamed"https://zbmath.org/authors/?q=ai:knifda.mohamedSummary: This study investigates the double-phase system with infinite potentials that vanish and convex-concave nonlinearity in Sobolev spaces with variable exponents on complete manifolds. Using the Nehari manifold and variational techniques, we aim to demonstrate the existence of at least two nonnegative nontrivial solutions. Our findings shed light on the behavior of this complex system and contribute to the mathematical analysis community's comprehension of nonlinear dynamics.Analysis on noncompact manifolds and index theory: Fredholm conditions and pseudodifferential operatorshttps://zbmath.org/1540.351722024-09-13T18:40:28.020319Z"Beschastnyi, Ivan"https://zbmath.org/authors/?q=ai:beschastnyi.ivan-yu"Carvalho, Catarina"https://zbmath.org/authors/?q=ai:carvalho.catarina-c"Nistor, Victor"https://zbmath.org/authors/?q=ai:nistor.victor"Qiao, Yu"https://zbmath.org/authors/?q=ai:qiao.yuSummary: We provide Fredholm conditions for compatible differential operators on certain Lie manifolds (that is, on certain possibly non-compact manifolds with nice ends). We discuss in more detail the case of manifolds with cylindrical, hyperbolic, and Euclidean ends, which are all covered by particular instances of our results. We also discuss applications to Schrödinger operators with singularities of the form \(r^{-2\gamma}\), \(\gamma \in \mathbb{R}_+\).
For the entire collection see [Zbl 1537.35003].Quasilinear elliptic systems involving the 1-Laplacian operator with subcritical and critical nonlinearitieshttps://zbmath.org/1540.351732024-09-13T18:40:28.020319Z"Carranza, Yino B. Cueva"https://zbmath.org/authors/?q=ai:carranza.yino-b-cueva"Pimenta, Marcos T. O."https://zbmath.org/authors/?q=ai:pimenta.marcos-t-oSummary: In this paper, we study some systems of elliptic PDEs involving the \(1\)-Laplacian operator. In the first one, we deal with the subcritical regime, while in the second, we study a system with nonlinearities with critical growth. The approach is based on an approximation argument, in which the solutions are obtained as the limit of related problems with the \(p\)-Laplacian operator. In order to overcome the lack of compactness in the critical case, a version of the Concentration of Compactness Principle of Lions is proved.Steady states of a diffusive population-toxicant model with negative toxicant-taxishttps://zbmath.org/1540.351742024-09-13T18:40:28.020319Z"Chu, Jiawei"https://zbmath.org/authors/?q=ai:chu.jiaweiSummary: This paper is dedicated to studying the steady state problem of a population-toxicant model with negative toxicant-taxis, subject to homogeneous Neumann boundary conditions. The model captures the phenomenon in which the population migrates away from regions with high toxicant density towards areas with lower toxicant concentration. This paper establishes sufficient conditions for the non-existence and existence of non-constant positive steady state solutions. The results indicate that in the case of a small toxicant input rate, a strong toxicant-taxis mechanism promotes population persistence and engenders spatially heterogeneous coexistence (see, Theorem 2.3). Moreover, when the toxicant input rate is relatively high, the results unequivocally demonstrate that the combination of a strong toxicant-taxis mechanism and a high natural growth rate of the population fosters population persistence, which is also characterized by spatial heterogeneity (see, Theorem 2.4).Ergodic mean field games: existence of local minimizers up to the Sobolev critical casehttps://zbmath.org/1540.351752024-09-13T18:40:28.020319Z"Cirant, Marco"https://zbmath.org/authors/?q=ai:cirant.marco"Cosenza, Alessandro"https://zbmath.org/authors/?q=ai:cosenza.alessandro"Verzini, Gianmaria"https://zbmath.org/authors/?q=ai:verzini.gianmariaSummary: We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.On a critical elliptic system with concave-convex nonlinearitieshttps://zbmath.org/1540.351762024-09-13T18:40:28.020319Z"Echarghaoui, Rachid"https://zbmath.org/authors/?q=ai:echarghaoui.rachid"Zaimi, Zakaria"https://zbmath.org/authors/?q=ai:zaimi.zakariaSummary: In this paper, we consider the following elliptic system involving critical Sobolev and Hardy-Sobolev exponents
\[
\begin{cases}
-\Delta u=\frac{2\alpha}{2^*}\vert u\vert^{\alpha -2} u\vert v\vert^{\beta}+\frac{\vert u\vert^{2^*(s)-2} u}{\vert x\vert^s}+a(x)\vert u\vert^{q-2} u & \text{in } \Omega, \\
-\Delta v=\frac{2\beta}{2^*}\vert u\vert^{\alpha} \vert v\vert^{\beta -2} v+\frac{\vert v\vert^{2^*(s)-2}v}{\vert x\vert^s}+b(x)\vert v \vert^{q-2} v & \text{in } \Omega, \\
u=v=0 & \text{on } \partial \Omega,
\end{cases}
\]
where \(\Omega\) is a bounded domains in \(\mathbb{R}^N\) satisfying some geometric condition, \(N \geq 3\), \(0<s<2\), \(1<q<2\), \(2^*(s):=\frac{2(N-s)}{(N-2)}\) and \(\alpha, \beta >1\) such that \(\alpha +\beta =2^* :=2^* (0)\). Our main result asserts that, if \(N>\max (4,\lfloor 2\,s\rfloor +2)\), then our problem has two disjoint and infinite sets of solutions. The present work may be seen as a positive answer to one open problem proposed by \textit{A. Ambrosetti} et al. [J. Funct. Anal. 122, No. 2, 519--543 (1994; Zbl 0805.35028)] for an elliptic systems.Nonradial solutions for coupled elliptic system with critical exponent in exterior domainhttps://zbmath.org/1540.351772024-09-13T18:40:28.020319Z"Guo, Yuxia"https://zbmath.org/authors/?q=ai:guo.yuxia"Li, Dewei"https://zbmath.org/authors/?q=ai:li.deweiSummary: We consider the following coupled Schrödinger system with critical exponent in \(\mathbb{R}^3\):
\[
\begin{cases}
-\Delta u+\lambda V(|y|)u = K_1(|y|)u^5+u^2v^3,\qquad & \text{ in } \mathbb{R}^3\backslash B_\epsilon(0),\\
-\Delta v+\lambda V(|y|)v = K_2(|y|)v^5+v^2u^3, \qquad & \text{ in } \mathbb{R}^3\backslash B_\epsilon(0),\\
u>0, v>0, \quad & \text{ in } \mathbb{R}^3\backslash B_\epsilon(0), \\
(u,v) = (0,0), \quad & \text{ on } \partial B_\epsilon (0), \\
u,v\in D^{1,2}(\mathbb{R}^3\backslash B_\epsilon(0))),
\end{cases}
\]
where \(V(|y|)\) is the potential function satifying \(0<V(|y|)\leq C\frac{1}{(1+|y|)^4}\) in \(\mathbb{R}^3\backslash B_\epsilon(0)\), \(\lambda >0\) is a constant. \( K_i\) (\( i = 1,2\)) are smooth bounded functions satisfying some suitable assumptions. \( B_\epsilon(0)\) is the ball centered at the origin with radius \(\epsilon\). By using Schmidt reduction arguments combine with the energy expansion and the critical point theory, we prove the existence of infinitely nonradial solutions for the system.A priori estimates and existence of solutions to a system of nonlinear elliptic equationshttps://zbmath.org/1540.351782024-09-13T18:40:28.020319Z"Jiang, Yongsheng"https://zbmath.org/authors/?q=ai:jiang.yongsheng"Wei, Na"https://zbmath.org/authors/?q=ai:wei.na"Wu, Yonghong"https://zbmath.org/authors/?q=ai:wu.yonghong.1Summary: We consider the following nonlinear elliptic equations
\[
\begin{cases}
- \Delta u + u + \lambda \phi (x) | u |^{r - 2} u = | u |^{p - 1} u, \, x \in \Omega,\\
- \Delta \phi (x) = | u |^q, \, x \in \Omega,\\
\phi (x) = u (x) = 0, \, x \in \partial \Omega,
\end{cases}
\tag{\(P\)}
\]
where \(p\), \(q\), \(r > 1\), \(\lambda\) is a parameter and \(\Omega \subset \mathbb{R}^3\) is a bounded domain. For \(q = r = 2\), the equations reduce to the Schrödinger-Poisson equations. Without the need of imposing constraint that \(q\) must be equal to \(r\), we establish a priori estimates, the nonexistence and existence of solutions for problem (P). Our results extend previous work for the case \(q = r\) to more general case.On a class of strongly coupled singular \((p, q)\)-Kirchhoff type systemshttps://zbmath.org/1540.351792024-09-13T18:40:28.020319Z"Rasouli, Seyyed Hashem"https://zbmath.org/authors/?q=ai:rasouli.seyyed-hashemSummary: The existence of positive solutions for a singular \((p, q)\)-Kirchhoff type system under Dirichlet boundary condition is studied. The main novelties consist in the presence of a Kirchhoff type system and in the strongly coupled reaction terms which tend to \(-\infty\). Our approach relies on the method of sub- and supersolutions.Generalized Poisson formula for second order elliptic systemshttps://zbmath.org/1540.351802024-09-13T18:40:28.020319Z"Soldatov, A. P."https://zbmath.org/authors/?q=ai:soldatov.aleksandr-pavlovichSummary: In the unit disc, we consider the Dirichlet problem for second order elliptic system with only higher order constant coefficients. We establish the unique solvability of this problem under the assumption that the problem is Fredholm, and obtain an explicit formula for the solution.A Fourier-Legendre spectral method for approximating the minimizers of \(\sigma_{2,p}\)-energyhttps://zbmath.org/1540.351812024-09-13T18:40:28.020319Z"Taghavi, M."https://zbmath.org/authors/?q=ai:taghavi.mojgan"Shahrokhi-Dehkordi, M. S."https://zbmath.org/authors/?q=ai:shahrokhi-dehkordi.m-sSummary: This paper proposes a Fourier-Legendre spectral method to find the minimizers of a variational problem, called \(\sigma_{2,p}\)-energy, in polar coordinates. Let \(\mathbb{X}\subset\mathbb{R}^n\) be a bounded Lipschitz domain and consider the energy functional \((1.1)\) whose integrand is defined by \(\mathbf{W}(\nabla u(x)):=(\sigma_2(u))^{\frac{p}{2}}+\Phi(\det\nabla u)\) over an appropriate space of admissible maps, \(\mathcal{A}_p(\mathbb{X})\). Using Fourier and Legendre interpolation errors, we obtain an error estimate for the energy functional and prove a convergence theorem for the proposed method. Furthermore, we apply the gradient descent method to solve a nonlinear algebraic system which is obtained by discretizing the Euler-Lagrange equations. The numerical experiments are performed to demonstrate the accuracy and effectiveness of our method.Symmetry breaking and multiplicity for supercritical elliptic Hamiltonian systems in exterior domainshttps://zbmath.org/1540.351822024-09-13T18:40:28.020319Z"Temgoua, Remi Yvant"https://zbmath.org/authors/?q=ai:temgoua.remi-yvantSummary: We consider positive solutions of the following elliptic Hamiltonian systems
\[
\begin{cases}
- \Delta u + u = a (x) v^{p - 1} \quad & \text{in } A_R \\
- \Delta v + v = b (x) u^{q - 1} & \text{in } A_R \\
\qquad\; u, v > 0 & \text{in } A_R \\
\quad\;\, u = v = 0 & \text{on } \partial A_R ,
\end{cases}
\tag{0.1}
\]
where \(A_R = \{x \in \mathbb{R}^N : | x | > R\}\), \(R > 0\), \(N > 3\), and \(a(x)\) and \(b(x)\) are positive continuous functions. Under certain symmetry and monotonicity properties on \(a(x)\) and \(b(x)\), we prove that (0.1) has a positive solution for \((p, q)\) above the standard critical hyperbola, that is, \(\frac{1}{p} + \frac{1}{q} < 1 - \frac{2}{N}\), enjoying the same symmetry and monotonicity properties as the weights \(a\) and \(b\). In the case when \(a(x) = b(x) = 1\), our result ensures multiplicity as it provides \(\left\lfloor \frac{N}{2} \right\rfloor - 1\) (being \(\lfloor \frac{N}{2} \rfloor\) the floor of \(\frac{N}{2}\)) non-radial positive solutions provided that
\[
(p - 1)(q - 1) > \Big(1 + \frac{2 N}{\Lambda_H}\Big)^2 \Big(\frac{q}{p}\Big),
\tag{0.2}
\]
where \(\Lambda_H\) is the optimal constant in Hardy inequality for the domain \(A_R\).Ground state and bounded state solutions for a critical stationary Maxwell system arising in electromagnetismhttps://zbmath.org/1540.351832024-09-13T18:40:28.020319Z"Xiang, Mingqi"https://zbmath.org/authors/?q=ai:xiang.mingqi"Chen, Linlin"https://zbmath.org/authors/?q=ai:chen.linlin"Yang, Miaomiao"https://zbmath.org/authors/?q=ai:yang.miaomiaoSummary: The aim of this paper is to establish the existence and multiplicity of bounded state solutions for a critical \((p(x), q(x))\)-curl system which arises in electromagnetism. The existence of nontrivial bounded state solutions is first obtained by applying the mountain pass lemma. Then the existence of ground state solutions is studied by restricting the analysis to the Nehari manifold. Furthermore, under some suitable assumptions, we prove that the mountain pass solution is actually a ground state solution. Finally, the existence of infinitely many solutions is investigated by using the genus theory combined with a truncated argument. The results obtained in this paper develop and complement several contributions concerning the \(p\)-curl operator and we focus on new existence results which are due to the presence of nonhomogeneous \((p(x), q(x))\)-curl operator and critical nonlinearity. To the best of our knowledge, our results are new even in the semilinear case.Almost sharp weighted Sobolev trace inequalities in the unit ball under constraintshttps://zbmath.org/1540.351842024-09-13T18:40:28.020319Z"An, Jiaxing"https://zbmath.org/authors/?q=ai:an.jiaxing"Dou, Jingbo"https://zbmath.org/authors/?q=ai:dou.jingbo"Han, Yazhou"https://zbmath.org/authors/?q=ai:han.yazhouSummary: In this paper, we establish some improved weighted Sobolev trace inequalities \(H^1(\rho^{1-2\sigma},\mathbb{B}^{n+1})\hookrightarrow L^q(\mathbb{S}^n)\) under the zero higher order moments constraint via the concentration compactness principle, where \(\rho\) is a defining function of \(\mathbb{B}^{n+1}\) and \(\sigma\in(0,1)\). This relates to the fractional (conformal) Laplacians and related problems in conformal geometry. We construct some test functions and show that the inequality is almost optimal when \(\sigma\in(0,\frac{1}{2}]\).Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured ballshttps://zbmath.org/1540.351852024-09-13T18:40:28.020319Z"Birindelli, Isabeau"https://zbmath.org/authors/?q=ai:birindelli.isabeau"Demengel, Françoise"https://zbmath.org/authors/?q=ai:demengel.francoise"Leoni, Fabiana"https://zbmath.org/authors/?q=ai:leoni.fabianaSummary: This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions \((\overline{\lambda}_\gamma,u_\gamma)\) of the equation
\[
F(D^2 u_\gamma)+\overline{\lambda}_\gamma\frac{u_\gamma}{r^\gamma}=0\text{ in }B(0,1)\smallsetminus \{0\},\quad u_\gamma=0\text{ on }\partial B(0,1)
\]
where \(u_\gamma>0\) in \(B(0,1)\smallsetminus \{0\}\) and \(\gamma>0\). We prove existence of radial solutions which are continuous on \(\overline{B(0,1)}\) in the case \(\gamma<2\), existence of unbounded solutions in the case \(\gamma=2\) and a non existence result for \(\gamma>2\). We also give, in the case of Pucci's operators, the explicit value of \(\overline{\lambda}_2\), which generalizes the Hardy-Sobolev constant for the Laplacian.Schwarz Lemma type estimates for solutions to nonlinear Beltrami equationhttps://zbmath.org/1540.351862024-09-13T18:40:28.020319Z"Klishchuk, Bogdan"https://zbmath.org/authors/?q=ai:klishchuk.bogdan-anatolevich"Salimov, Ruslan"https://zbmath.org/authors/?q=ai:salimov.ruslan-radikovich"Stefanchuk, Mariia"https://zbmath.org/authors/?q=ai:stefanchuk.mariia-vSummary: We continue to investigate the regular homeomorphic solutions to nonlinear Beltrami equation introduced in [\textit{A. Golberg} et al.,
Complex Var. Elliptic Equ. 65, No. 1, 6--21 (2020; Zbl 1427.30037)]. Schwarz Lemma type estimates are obtained involving the length-area method. The lower bounds for the inverses are also established.
For the entire collection see [Zbl 1531.35008].On the exterior Dirichlet problem for Hessian-type fully nonlinear elliptic equationshttps://zbmath.org/1540.351872024-09-13T18:40:28.020319Z"Li, Xiaoliang"https://zbmath.org/authors/?q=ai:li.xiaoliang"Wang, Cong"https://zbmath.org/authors/?q=ai:wang.cong.4Summary: We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form
\[
f(\lambda( D^2u))=g(x)
\]
with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by \textit{L. Caffarelli} et al. [Acta Math. 155, 261--301 (1985; Zbl 0654.35031)], \textit{N. S. Trudinger} [Acta Math. 175, No. 2, 151--164 (1995; Zbl 0887.35061)] and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that \(f\) is a concave function. In this paper, based on Perron's method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming \(f\) to satisfy certain structure conditions as in [Caffarelli et al., loc. cit.] and [Trudinger, loc. cit.] but without requiring the concavity of \(f\). The equations in our setting may embrace the well-known Monge-Ampère equations, Hessian equations and Hessian quotient equations as special cases.Ground state normalized solutions to the Kirchhoff equation with general nonlinearities: mass supercritical casehttps://zbmath.org/1540.351882024-09-13T18:40:28.020319Z"Wang, Qun"https://zbmath.org/authors/?q=ai:wang.qun"Qian, Aixia"https://zbmath.org/authors/?q=ai:qian.aixia(no abstract)On the large solutions to a class of \(k\)-Hessian problemshttps://zbmath.org/1540.351892024-09-13T18:40:28.020319Z"Wan, Haitao"https://zbmath.org/authors/?q=ai:wan.haitaoSummary: In this paper, we consider the \(k\)-Hessian problem \(S_k (D^2 u) = b(x)f(u)\) in \(\Omega, u = +\infty\) on \(\partial\Omega\), where \(\Omega\) is a \(C^{\infty}\)-smooth bounded strictly \((k-1)\)-convex domain in \(\mathbb{R}^N\) with \(N \geq 2\), \(b \in C^{\infty}(\Omega)\) is positive in \(\Omega\) and may be singular or vanish on \(\partial\Omega\), \(f \in C[0, \infty) \cap C^1 (0, \infty)\) (or \(f\in C^1 (\mathbb{R}))\) is a positive and increasing function. We establish the first expansions (equalities) of \(k\)-convex solutions to the above problem when \(f\) is borderline regularly varying and \(\Gamma\)-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of \(f\) and principal curvatures of \(\partial \Omega\) on the first expansion of solutions. For the latter, we find the principal curvatures of \(\partial \Omega\) have no influences on the expansions. Our results and methods are quite different from the existing ones (including \(k=N)\). Moreover, we know the existence of \(k\)-convex solutions to the above problem (including \(k=N)\) is still an open problem when \(b\) possesses high singularity on \(\partial\Omega\) and \(f\) satisfies Keller-Osserman type condition. For the radially symmetric case in the ball, we give a positive answer to this open problem, and then we further show the global estimates for all radial large solutions.The iterative properties of solutions for a singular \(k\)-Hessian systemhttps://zbmath.org/1540.351902024-09-13T18:40:28.020319Z"Zhang, Xinguang"https://zbmath.org/authors/?q=ai:zhang.xinguang"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng.4|chen.peng.1|chen.peng.2"Wu, Yonghong"https://zbmath.org/authors/?q=ai:wu.yonghong.1"Wiwatanapataphee, Benchawan"https://zbmath.org/authors/?q=ai:wiwatanapataphee.benchawanSummary: In this paper, we focus on the uniqueness and iterative properties of solutions for a singular \(k\)-Hessian system involving coupled nonlinear terms with different properties. Unlike the existing work, instead of directly dealing with the system, we use a coupled technique to transfer the Hessian system to an integral equation, and then by introducing an iterative technique, the iterative properties of solution are derived including the uniqueness of solution, iterative sequence, the error estimation and the convergence rate as well as entire asymptotic behaviour.Existence and multiplicity of solutions for a locally coercive elliptic equationhttps://zbmath.org/1540.351912024-09-13T18:40:28.020319Z"Arcoya, David"https://zbmath.org/authors/?q=ai:arcoya.david"de Paiva, Francisco Odair"https://zbmath.org/authors/?q=ai:de-paiva.francisco-odair"Mendoza, José M."https://zbmath.org/authors/?q=ai:mendoza.jose-mSummary: For a bounded domain \(\Omega \), we establish existence and multiplicity of nontrivial solutions for the semilinear elliptic problem
\[
\begin{cases}
\begin{alignedat}{3}
-\Delta u & = {g(u)} - h(x) f(u), & \text{ in } & \Omega \\
u & = 0, & \text{ on } & \partial \Omega,
\end{alignedat}
\end{cases}
\]
where \(h\in L^\infty (\Omega)\) is nonnegative and nontrivial, \(g\) is asymptotically linear, \(f\) is superlinear and \({g(0)}=f(0)=0\). We also study the existence of solutions for the problem
\[
\begin{cases}
\begin{alignedat}{3}
-\Delta u & = {g(u)} - h(x)f(u)+k(x), &\text{ in } & \Omega \\
u & = 0, & \text{ on } & \partial \Omega,
\end{alignedat}
\end{cases}
\]
when \(k\in L^2(\Omega)\).On the fractional Musielak-Sobolev spaces in \(\mathbb{R}^d\): embedding results \& applicationshttps://zbmath.org/1540.351922024-09-13T18:40:28.020319Z"Bahrouni, Anouar"https://zbmath.org/authors/?q=ai:bahrouni.anouar"Missaoui, Hlel"https://zbmath.org/authors/?q=ai:missaoui.hlel"Ounaies, Hichem"https://zbmath.org/authors/?q=ai:ounaies.hichemSummary: This paper deals with new continuous and compact embedding theorems for the fractional Musielak-Sobolev spaces in \(\mathbb{R}^d\). As an application, using the variational methods, we obtain the existence of a nontrivial weak solution for the following Schrödinger equation
\[
(- \Delta)_{g_{x, y}}^s u + V(x) g(x, x, u) = b(x) |u|^{p(x) - 2} u, \text{ for all } x \in \mathbb{R}^d,
\]
where \((- \Delta)_{g_{x, y}}^s\) is the fractional Museilak \(g_{x, y}\)-Laplacian, \(V\) is a potential function, \(b \in L^{\delta^\prime (x)}(\mathbb{R}^d)\), and \(p, \delta \in C \big(\mathbb{R}^d, (1, + \infty)\big) \cap L^\infty (\mathbb{R}^d)\). We would like to mention that the theory of the fractional Musielak-Sobolev spaces is in a developing state and there are few papers on this topic, see [\textit{J. C. de Albuquerque} et al., J. Geom. Anal. 33, No. 4, Paper No. 130, 37 p. (2023; Zbl 1518.46021);
\textit{E. Azroul} et al., Appl. Anal. 101, No. 6, 1933--1952 (2022; Zbl 1497.46040); Appl. Anal. 102, No. 1, 195--219 (2023; Zbl 1523.46024)]. Note that, all these latter works dealt with the bounded case and there are no results devoted for the fractional Musielak-Sobolev spaces in \(\mathbb{R}^d\). Since the embedding results are crucial in applying variational methods, this work will provide a bridge between the fractional Mueislak-Sobolev theory and PDE's.Positive solutions to semilinear Dirichlet problems with general boundary datahttps://zbmath.org/1540.351932024-09-13T18:40:28.020319Z"Beznea, Lucian"https://zbmath.org/authors/?q=ai:beznea.lucian"Teodor, Alexandra"https://zbmath.org/authors/?q=ai:teodor.alexandraSummary: We give a probabilistic representation of the solution to a semilinear elliptic Dirichlet problem with general (discontinuous) boundary data. The boundary behaviour of the solution is in the sense of the controlled convergence initiated by \textit{A. Cornea} [C. R. Acad. Sci., Paris, Sér. I 320, No. 2, 159--164 (1995; Zbl 0828.31003); in: Analysis and topology. A volume dedicated to the memory of S. Stoilow. Singapore: World Scientific. 257-275 (1998; Zbl 0940.31007)]. Uniqueness results for the solution are also provided.Regularity results to a class of elliptic equations with explicit \(u\)-dependence and Orlicz growthhttps://zbmath.org/1540.351942024-09-13T18:40:28.020319Z"Capone, Claudia"https://zbmath.org/authors/?q=ai:capone.claudia"Passarelli di Napoli, Antonia"https://zbmath.org/authors/?q=ai:passarelli-di-napoli.antoniaSummary: We study the regularity properties of the weak solutions \(u:\Omega \subseteq \mathbb{R}^n\to\mathbb{R}\) to problems of the type
\[
\begin{cases}
-\mathrm{div}\, a(x,Du)+b(x)\phi'(|u|) \frac{u}{|u|} =f & \text{in }\Omega \\
u=0 & \text{on }\partial\Omega
\end{cases}
\]
with \(\Omega\subset\mathbb{R}^n\) a bounded open set and where the function \(a(x,\xi)\) satisfies growth conditions with respect to the second variable expressed through an N-function \(\phi\). We prove that, under a suitable interplay between the lower order terms and the datum \(f\), which is assumed only to belong to \(L^1 (\Omega)\), the solutions are bounded in \(\Omega\). Next, if \(a(x,\xi)\) depends on \(x\) through a Hölder continuous function, we take advantage from the boundedness of the solution \(u\) to prove the higher differentiability and the higher integrability of its gradient, under mild assumptions on the data.An upper bound for the least energy of a sign-changing solution to a zero mass problemhttps://zbmath.org/1540.351952024-09-13T18:40:28.020319Z"Clapp, Mónica"https://zbmath.org/authors/?q=ai:clapp.monica"Maia, Liliane"https://zbmath.org/authors/?q=ai:maia.liliane-a"Pellacci, Benedetta"https://zbmath.org/authors/?q=ai:pellacci.benedettaThis article is concerned with qualitative properties of sign-changing solution to the semilinear elliptic equation \(-\Delta u=f(u)\) in \(\mathcal{D}^{1,2}(\mathbb{R}^N)\), \(N\geq 5\). The nonlinearity \(f\) is assumed to be subcritical at infinity and supercritical near the origin. Let \(c_0\) be the the ground state energy of the above equation. The main result of the article establishes the existence of a nonradial sign-changing solution \(\widehat \omega\) such that its energy satisfies the double sided inequality
\[
2c_0<\frac{1}{2}\int_{\mathbb{R}^N} |\nabla \widehat \omega |^2-\int_{\mathbb{R}^N} F(\widehat \omega)<\begin{cases} 12 c_0 &\mbox{if } N=5,6\\
10c_0 &\mbox{if } N\geq 7. \end{cases}
\]
The approach combines the study of a special finite group of symmetries with the use of the concentration compacntess principle.
Reviewer: Marius Ghergu (Dublin)Normalized solutions for Schrödinger-Poisson equation with prescribed mass: the Sobolev subcritical case and the Sobolev critical case with mixed dispersionhttps://zbmath.org/1540.351962024-09-13T18:40:28.020319Z"Hu, Die"https://zbmath.org/authors/?q=ai:hu.die"Tang, Xianhua"https://zbmath.org/authors/?q=ai:tang.xian-hua"Jin, Peng"https://zbmath.org/authors/?q=ai:jin.pengSummary: In this paper, we study the following Schrödinger Poisson equation
\[
\begin{cases}
-\Delta u -\lambda u -\gamma (|x|^{-1} \ast |u|^2) u = f(u), & x \in \mathbb{R}^3; \\
\displaystyle\int\limits_{\mathbb{R}^3} u^2 dx = c,
\end{cases}
\]
where \(c>0\), \(\lambda \in \mathbb{R}\) and \(f \in \mathcal{C}(\mathbb{R}, \mathbb{R})\). When \(\gamma <0\) and \(f\) satisfies some weaker \(L^2\)-supercritical conditions in Sobolev subcritical case, we show the existence of normalized solutions. When \(f(u) = \mu |u|^{q-2} u + |u|^4 u\) with \(\mu >0\), \(\gamma >0\) and \(2 < q < \frac{10}{3}\), a more complicated situation, we obtain the existence of multiple solutions. Our main tools are some technically simpler than the \textit{N. Ghoussoub} minimax principle [Duality and perturbation methods in critical point theory. Cambridge: Cambridge University Press (1993; Zbl 0790.58002)], which not only allow us to construct a bounded (PS) sequence under some weaker conditions on \(f\) than before, but help to yield richer results. In particular, our results generalize and improve some ones in [\textit{L. Jeanjean} and \textit{T. T. Le}, J. Differ. Equations 303, 277--325 (2021; Zbl 1475.35163)] and some other related literatures.A nonlinear elliptic equation with a degenerate diffusion and a source term in \(L^1\)https://zbmath.org/1540.351972024-09-13T18:40:28.020319Z"Leloup, Guillaume"https://zbmath.org/authors/?q=ai:leloup.guillaume"Lewandowski, Roger"https://zbmath.org/authors/?q=ai:lewandowski.rogerSummary: We study a simplified equation governing turbulent kinetic energy \(k\) in a bounded domain, arising from turbulence modeling where the eddy diffusion is given by \(\varrho (x) + \varepsilon\), with \(\varrho\) representing the Prandtl mixing length of the order of the distance to the boundary, and a right-hand side in \(L^1\). We obtain estimates of \(\sqrt{\varrho} \nabla k\) in \(L^q\) spaces and we establish the convergence toward the formal limit equation in the sense of the distributions as \(\varepsilon\) goes to 0.Infinitely many solutions of strongly degenerate Schrödinger elliptic equations with vanishing potentialshttps://zbmath.org/1540.351982024-09-13T18:40:28.020319Z"My, Bui Kim"https://zbmath.org/authors/?q=ai:my.bui-kimSummary: In this paper, we are concerned with the existence of infinitely many nontrivial solutions to the following semilinear degenerate elliptic equation
\[
-\Delta_\lambda u + V(x) u = f(x, u) \quad\text{in }\mathbb{R}^N, N \geq 3,
\]
where \(V: \mathbb{R}^N\rightarrow\mathbb{R}\) is a potential function and allowed to be vanishing at infinitely, \(f: \mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}\) is a given function and \(\Delta_\lambda\) is the strongly degenerate elliptic operator. Under suitable assumptions on the potential \(V\) and the nonlinearity \(f\), some results on the multiplicity of solutions are proved. The proofs are based on variational methods, in particular, on the well-known mountain pass lemma of Ambrosetti-Rabinowitz. Due to the vanishing potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results [Zbl 1263.35076; Zbl 1465.35234; Zbl 1433.35120; Zbl 1392.35146; Zbl 1364.35103; Zbl 1448.35131].New regularization and error estimate on terminal value problem for elliptic equationshttps://zbmath.org/1540.351992024-09-13T18:40:28.020319Z"Triet, Nguyen Anh"https://zbmath.org/authors/?q=ai:nguyen-anh-triet."Nam, Danh Hua Quoc"https://zbmath.org/authors/?q=ai:nam.danh-hua-quoc"Can, Nguyen Huu"https://zbmath.org/authors/?q=ai:can.nguyen-huu"Long, Le Dinh"https://zbmath.org/authors/?q=ai:long.le-dinhSummary: In this paper, we study the terminal value problem for elliptic equation with various forms of source functions. This equation has many applications in the fields of physics, mechanics, electromagnetism. Until now, there are not many studies that focus on the regularization in \(L^p\) spaces. Our results are the first work on the inverse problem for elliptic equations in \(L^p \). The principal technique is to use Fourier regularized solution combined with Sobolev embeddings. The error between the exact and regularized solutions are obtained in \(L^p\) under some suitable assumptions on the Cauchy data.Entropy solution of nonlinear elliptic \(p(u)\)-Laplacian problemhttps://zbmath.org/1540.352002024-09-13T18:40:28.020319Z"Abbassi, Adil"https://zbmath.org/authors/?q=ai:abbassi.adil"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakir"Temghart, Said Ait"https://zbmath.org/authors/?q=ai:ait-temghart.saidSummary: In this paper, we prove the existence of an entropy solution for a class of nonlinear nonlocal elliptic problem associated to the following equation
\[
\begin{cases}
\begin{aligned}
&b(u)-\mathrm{d}\mathrm{i}\mathrm{v}a(x,u,\nabla u)-\mathrm{d}\mathrm{i}\mathrm{v}\phi (u)=f &&\text{in } \Omega\\
&u=0 &&\text{on } \partial \Omega,
\end{aligned}
\end{cases}
\]
where the operator \(-\mathrm{d}\mathrm{i}\mathrm{v}a(x,u,\nabla u)\) is called \(p(u)\)-Laplacian.
For the entire collection see [Zbl 1515.35012].Existence and uniqueness of weak solution for a class of nonlinear degenerate elliptic problems in weighted Sobolev spaceshttps://zbmath.org/1540.352012024-09-13T18:40:28.020319Z"El Ouaarabi, Mohamed"https://zbmath.org/authors/?q=ai:el-ouaarabi.mohamed"Abbassi, Ail"https://zbmath.org/authors/?q=ai:abbassi.ail"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakirSummary: This work is devoted to study the existence and uniqueness of weak solution for a class of nonlinear degenerate elliptic problems of the following form
\[
-\mathrm{{div}}\Big [ \omega_1 \mathcal{A}(x,\nabla u)+\omega_2 \mathcal{B}(x,u,\nabla u)\Big]+ \omega_3\mathcal{H}(x,u)=\phi (x),
\]
where \(\omega_1\), \(\omega_2\) and \(\omega_3\) are \(A_p\)-weight functions and the operators \(\mathcal{A}\), \(\mathcal{B}\) and \(\mathcal{H}\) are Caratéodory functions that satisfy some conditions and the right-hand side term \(\phi \in L^{p'}(\varOmega ,\omega_1^{1-p'})\). Our technical approach is based on the Browder-Minty Theorem and the weighted Sobolev spaces theory.
For the entire collection see [Zbl 1515.35012].Normalized solutions to the Kirchhoff equation with triple critical exponents in \(\mathbb{R}^4\)https://zbmath.org/1540.352022024-09-13T18:40:28.020319Z"Fang, Xingling"https://zbmath.org/authors/?q=ai:fang.xingling"Ou, Zengqi"https://zbmath.org/authors/?q=ai:ou.zengqi"Lv, Ying"https://zbmath.org/authors/?q=ai:lv.yingSummary: In this paper, we investigate the normalized solutions for the nonlinear critical Kirchhoff equations with combined nonlinearities:
\[
\begin{cases}
\displaystyle - \left(a + b \int_{\mathbb{R}^N} | \nabla u |^2 d x\right) \Delta u = \lambda u + \mu | u | u + | u |^2 u,\\
\displaystyle \int_{\mathbb{R}^N} u^2 d x = c^2, \quad x \in \mathbb{R}^N,
\end{cases}
\]
where \(N = 4\), \(\lambda\), \(\mu \in \mathbb{R}\) and \(a\), \(b\), \(c > 0\) are constants. In \(\mathbb{R}^4\), some interesting phenomena occur, which are, the \(L^2\)-critical exponent for \(\Delta u\) is \(2 + \frac{4}{N} = 3\), while the \(L^2\)-critical exponent for \((\int_{\mathbb{R}^4} | \nabla u |^2 d x) \Delta u\) is equal to the Sobolev critical exponent, i.e., \(2 + \frac{8}{N} = \frac{2 N}{N - 2} = 4\). This paper investigates the case that the nonlinearity with triple critical term and proves the existence of a positive mountain-pass type normalized solution through variational methods and energy estimations.Existence and multiplicity of solutions for general quasi-linear elliptic equations with sub-cubic nonlinearitieshttps://zbmath.org/1540.352032024-09-13T18:40:28.020319Z"Huang, Chen"https://zbmath.org/authors/?q=ai:huang.chen"Zhang, Jianjun"https://zbmath.org/authors/?q=ai:zhang.jianjun"Zhong, Xuexiu"https://zbmath.org/authors/?q=ai:zhong.xuexiuSummary: We consider the following quasi-linear Schrödinger equation
\[
-\sum_{i, j=1}^N D_j (a_{ij}(u) D_i u) + \frac{1}{2} \sum_{i, j=1}^N D_s a_{ij}(u) D_i u D_j u + V(x)u = f(u),\; x \in \mathbb{R}^N, N \geq 3,
\]
which includes the modified nonlinear Schrödinger equations. Combining a \(p\)-Laplacian perturbation argument, we obtain the existence of positive solutions to the problem above and multiple solutions with additional symmetry conditions on \(a_{ij}\) and \(f\). A new perturbation approach is used to treat the sub-cubic nonlinearity. In particular, for \(f(u) = |u|^{p-2} u\), the case \(p \in (2, 4)\) is also considered.Least energy sign-changing solution for \(N\)-Kirchhoff problems with logarithmic and exponential nonlinearitieshttps://zbmath.org/1540.352042024-09-13T18:40:28.020319Z"Huang, Ting"https://zbmath.org/authors/?q=ai:huang.ting"Shang, Yan-Ying"https://zbmath.org/authors/?q=ai:shang.yanyingSummary: In this paper, we are concerned with the existence of least energy sign-changing solutions for the following \(N\)-Laplacian Kirchhoff-type problem with logarithmic and exponential nonlinearities:
\[
\begin{cases}
-\left(a + b\int_\Omega|\nabla u|^N dx\right) \Delta_N u = |u|^{p-2}u\ln|u|^2 + \lambda f(u), &\text{in }\Omega,\\
u=0, &\text{on }\partial\Omega,
\end{cases}
\]
where \(f(t)\) behaves like \(\exp\left(\alpha|t|^{\frac{N}{N - 1}}\right)\). Combining constrained variational method, topological degree theory and quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution \(u_b\) with precisely two nodal domains. Moreover, we show that the energy of \(u_b\) is strictly larger than two times of the ground state energy and analyze the convergence property of \(u_b\) as \(b\searrow 0\).Infinitely many small energy solutions to nonlinear Kirchhoff-Schrödinger equations with the \(p\)-Laplacianhttps://zbmath.org/1540.352052024-09-13T18:40:28.020319Z"Kim, In Hyoun"https://zbmath.org/authors/?q=ai:kim.in-hyoun"Kim, Yun-Ho"https://zbmath.org/authors/?q=ai:kim.yunhoSummary: This paper is devoted to deriving the multiplicity result of solutions to the nonlinear elliptic equations of Kirchhoff-Schrödinger type on a class of a nonlocal Kirchhoff coefficient which slightly differs from the previous related works. More precisely, the main purpose of this paper, under the various conditions for a nonlinear term, is to show that our problem has a sequence of infinitely many small energy solutions. In order to obtain such a multiplicity result, the dual fountain theorem is used as the primary tool.Existence of positive solutions for a \(p\)-Schrödinger-Kirchhoff integro-differential equation with critical growthhttps://zbmath.org/1540.352062024-09-13T18:40:28.020319Z"Mayorga-Zambrano, Juan"https://zbmath.org/authors/?q=ai:mayorga-zambrano.juan"Cumbal-López, Henry"https://zbmath.org/authors/?q=ai:cumbal-lopez.henrySummary: We consider the \(p\)-Schrödinger-Kirchhoff-type equation
\[
-\left[\varepsilon^pa+b \varepsilon^\beta\left(\int_{\mathbb{R}^N}|\nabla v|^pdx\right)^{p-1}\right]\Delta_pv+M(x)|v|^{p-2}v=\tilde{\sigma}(v),
\tag{\(\mathrm{P}_{\varepsilon}\)}
\]
for \(v\in{\mathrm{W}}^{1,p}({\mathbb{R}}^{\mathrm{N}})\), where \(\tilde{\sigma}(s) = \lambda f(s) + |s|^{p^*-2} s\), \(b\ge 0\), \(a,\varepsilon ,\lambda >0\), \(\beta =p^2-Np+N\) and \(1<p<N\le p+1<p^*-2\), \(p^*=pN/(N-p)\). We assume that \(M\) and \(f\) verify conditions like those considered by \textit{J. Wang} et al. [J. Differ. Equations 253, No. 7, 2314--2351 (2012; Zbl 1402.35119)]; in particular, \({\mathcal{M}}=\{x\in{\mathbb{R}}^{\mathrm{N}} \, / \, M(x) = M_0\}\ne \emptyset \), \(M_0 = \inf M>0\). Thanks to a study of the ground state of the limit problem associated to \(( \mathrm{P}_{\varepsilon})\), we prove, by the method of Nehari manifold, the existence of a positive ground state of \(( \mathrm{P}_{\varepsilon})\). By a Ljusternik-Schnirelmann scheme it's shown, for \(\varepsilon\) small and \(\lambda\) big, that \(( \mathrm{P}_{\varepsilon})\) has at least \({\mathrm{cat}}({\mathcal{M}},{\mathcal{M}}_\delta)\) positive solutions, where \({\mathcal{M}}_\delta = \{x\in{\mathbb{R}}^{\mathrm{N}} \, / \, {\mathrm{dist}}(x,{\mathcal{M}})<\delta \}\), \(\delta >0\).Dirichlet problems with anisotropic principal part involving unbounded coefficientshttps://zbmath.org/1540.352072024-09-13T18:40:28.020319Z"Motreanu, Dumitru"https://zbmath.org/authors/?q=ai:motreanu.dumitru"Tornatore, Elisabetta"https://zbmath.org/authors/?q=ai:tornatore.elisabettaSummary: This article establishes the existence of solutions in a weak sense for a quasilinear Dirichlet problem exhibiting anisotropic differential operator with unbounded coefficients in the principal part and full dependence on the gradient in the lower order terms. A major part of this work focuses on the existence of a uniform bound for the solution set in the anisotropic setting. The unbounded coefficients are handled through an appropriate truncation and a priori estimates.Existence of solutions to quasilinear Schrödinger equations with exponential nonlinearityhttps://zbmath.org/1540.352082024-09-13T18:40:28.020319Z"Severo, Uberlandio B."https://zbmath.org/authors/?q=ai:severo.uberlandio-batista"Ribeiro, Bruno H. C."https://zbmath.org/authors/?q=ai:ribeiro.bruno-h-c"de S. Germano, Diogo"https://zbmath.org/authors/?q=ai:de-s-germano.diogoSummary: In this article we study the existence of solutions to quasilinear Schrödinger equations in the plane, involving a potential that can change sign and a nonlinear term that may be discontinuous and exhibit exponential critical growth. To prove our existence result, we combine the Trudinger-Moser inequality with a fixed point theorem.A planar Kirchhoff equation with exponential growth and double nonlocal termhttps://zbmath.org/1540.352092024-09-13T18:40:28.020319Z"Tordecilla, Jesus Alberto Leon"https://zbmath.org/authors/?q=ai:tordecilla.jesus-alberto-leonSummary: We investigate the existence of a positive solution for an elliptic problem of the Kirchhoff type involving a nonlocal operator. The nonlinearity considered in the equation combined a nonlocal term with an exponential growth governed by the Trudinger-Moser inequality.Entropy solution for equation with a measure valued potential in hyperbolic spacehttps://zbmath.org/1540.352102024-09-13T18:40:28.020319Z"Vil'danova, Venera F."https://zbmath.org/authors/?q=ai:vildanova.venera-fidarisovna"Mukminov, Farit Kh."https://zbmath.org/authors/?q=ai:mukminov.farit-khSummary: We consider the Dirichlet problem in the hyperbolic space for a nonlinear elliptic equation of the second order with singular measurevalued potential. The assumptions on the structure of the equation are stated in terms of a generalized \(N\)-function. It is shown that this problem has an entropy solution.Multiple solutions to a transmission problem with a critical Hardy-Sobolev exponential source termhttps://zbmath.org/1540.352112024-09-13T18:40:28.020319Z"Wang, Yue"https://zbmath.org/authors/?q=ai:wang.yue.8Summary: In the paper there are established many results for a transmission problem with critical Hardy-Sobolev exponential source term \(\frac{u^3}{|x|}\) in \(\mathbb{R}^3\). We obtain that there are at least three weakly nontrivial solutions when a positive coefficient of nonhomogeneous term is enough small using the variational method. Next infinitely many classical solutions are obtained when the coefficient equals to zero. Moreover, a new compactness condition is derived with the help of Brezis-Lieb's lemma and Mazur's lemma.Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditionshttps://zbmath.org/1540.352122024-09-13T18:40:28.020319Z"Mukherjee, Tuhina"https://zbmath.org/authors/?q=ai:mukherjee.tuhina"Pucci, Patrizia"https://zbmath.org/authors/?q=ai:pucci.patrizia"Sharma, Lovelesh"https://zbmath.org/authors/?q=ai:sharma.loveleshThe aim of this paper is to investigate the existence and multiplicity of solutions to some singular problems, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian. First the authors present the functional framework suitable for the spectral fractional Laplace operator with mixed boundary conditions. They also review the Caffarelli and Silvestre extension technique, which offers an alternative definition of the fractional Laplace operator using an auxiliary problem. Further, they analyze fiber maps and the Nehari manifold to prove the existence of minimizers over suitable subsets of it. As the main result, they show that the infimum of the energy functional over \(\mathcal{N}^+_\lambda\) and \(\mathcal{N}^-_\lambda\) are achieved and are the desired weak solutions. Regularity results are also established.
Reviewer: Said El Manouni (Riyadh)Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbationhttps://zbmath.org/1540.352132024-09-13T18:40:28.020319Z"Chen, Hua"https://zbmath.org/authors/?q=ai:chen.hua"Liao, Xin"https://zbmath.org/authors/?q=ai:liao.xin"Zhang, Ming"https://zbmath.org/authors/?q=ai:zhang.ming.5|zhang.ming.3Summary: In this paper, we investigate the existence of weak solutions for a class of degenerate elliptic Dirichlet problems with critical nonlinearity and a logarithmic perturbation, i.e.
\[
\begin{cases}
-(\Delta_x u + (\alpha +1)^2|x|^{2\alpha}\Delta_y u) = u^{\frac{Q+2}{Q-2}} + \lambda u\log u^2,\\
u = 0 \text{ on } \partial\Omega ,
\end{cases}\tag{0.2}
\]
where \((x, y)\in\Omega\subset\mathbb{R}^N = \mathbb{R}^m\times\mathbb{R}^n\) with \(m \geq1\), \(n \geq 0\), \(\Omega\cap\{x = 0\} \neq \emptyset\) is a bounded domain, the parameter \(\alpha \geq 0\) and \(Q = m + n(\alpha + 1)\) denotes the ``homogeneous dimension'' of \(\mathbb{R}^N\). When \(\lambda = 0\), we know that from [\textit{A. Kogoj} and \textit{E. Lanconelli}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 12, 4637--4649 (2012; Zbl 1260.35020)] the problem (0.2) has a Pohožaev-type non-existence result. Then for \(\lambda\in\mathbb{R}\setminus\{0\}\), we establish the existences of non-negative ground state weak solutions and non-trivial weak solutions subject to certain conditions.3D numerical simulation of an anisotropic bead type thermistor and multiplicity of solutionshttps://zbmath.org/1540.352142024-09-13T18:40:28.020319Z"Lahrache, Manar"https://zbmath.org/authors/?q=ai:lahrache.manar"Ortegón Gallego, Francisco"https://zbmath.org/authors/?q=ai:ortegon-gallego.francisco"Rhoudaf, Mohamed"https://zbmath.org/authors/?q=ai:rhoudaf.mohamed(no abstract)Degenerate Kolmogorov equations and ergodicity for the stochastic Allen-Cahn equation with logarithmic potentialhttps://zbmath.org/1540.352152024-09-13T18:40:28.020319Z"Scarpa, Luca"https://zbmath.org/authors/?q=ai:scarpa.luca"Zanella, Margherita"https://zbmath.org/authors/?q=ai:zanella.margheritaSummary: Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen-Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at the respective potential barriers, making thus the corresponding Kolmogorov equation not uniformly elliptic in space. First, existence and uniqueness of invariant measures and ergodicity are discussed. Then, classical solutions to some regularised Kolmogorov equations are explicitly constructed. Eventually, a sharp analysis of the blow-up rates of the regularised solutions and a passage to the limit with a specific scaling yield existence à la Friedrichs for the original Kolmogorov equation.Existence results for variational-hemivariational inequality systems with nonlinear couplingshttps://zbmath.org/1540.352162024-09-13T18:40:28.020319Z"Bai, Yunru"https://zbmath.org/authors/?q=ai:bai.yunru"Costea, Nicuşor"https://zbmath.org/authors/?q=ai:costea.nicusor"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengdaSummary: In this paper we investigate a system of coupled inequalities consisting of a variational-hemivariational inequality and a quasi-hemivariational inequality on Banach spaces. The approach is topological, and a wide variety of existence results is established for both bounded and unbounded constraint sets in real reflexive Banach spaces. Applications to Contact Mechanics are provided in the last section of the paper. More precisely, we consider a contact model with (possibly) multivalued constitutive law whose variational formulation leads to a coupled system of inequalities. The weak solvability of the problem is proved via employing the theoretical results obtained in the previous section. The novelty of our approach comes from the fact that we consider two potential contact zones and the variational formulation allows us to determine simultaneously the displacement field and the Cauchy stress tensor.Multi-piece of bubble solutions for a nonlinear critical elliptic equationhttps://zbmath.org/1540.352172024-09-13T18:40:28.020319Z"Du, Fan"https://zbmath.org/authors/?q=ai:du.fan"Hua, Qiaoqiao"https://zbmath.org/authors/?q=ai:hua.qiaoqiao"Wang, Chunhua"https://zbmath.org/authors/?q=ai:wang.chunhua"Wang, Qingfang"https://zbmath.org/authors/?q=ai:wang.qingfangSummary: We revisit the following nonlinear critical elliptic equation
\[
\Delta u + V(| y^\prime |, y'') u = u^{\frac{N + 2}{N - 2}}, \quad u > 0, \quad u \in H^1( \mathbb{R}^N),
\]
where \(( y^\prime, y'') \in \mathbb{R}^3 \times \mathbb{R}^{N - 3}\), \(V(| y^\prime |, y'')\) is a bounded non-negative function in \(\mathbb{R}^+ \times \mathbb{R}^{N - 3} \). Assuming that \(r^2 V(r, y'')\) has a stable critical point \(( r_0, y_0'')\) with \(r_0 > 0\) and \(V( r_0, y_0'') > 0\), by using a modified finite-dimensional reduction method and various local Pohozaev identities, we prove that the problem above has multi-piece of bubble solutions, whose energy can be made arbitrarily large. Since there involves a new variable \(\overline{h}\) in the concentrated points \(\{ x_j^\pm \}_{j = 1}^m\) during the reduction process, we have to obtain a more precise estimate for the error term. And the bubble solutions are centered at points lying on the top and the bottom circles of a cylinder. Particularly, in one of these cases, the bubble solutions can concentrate at a pair of symmetric points relative to the origin. Our results present a new clustering type of blow-up phenomenon and we think the reason why this phenomenon can occur is mainly because that the function \(r^2 V(r, y'')\) has non-isolated critical points.On the location of the concentration points of the two-dimensional Hénon equationhttps://zbmath.org/1540.352182024-09-13T18:40:28.020319Z"Du, Geyang"https://zbmath.org/authors/?q=ai:du.geyangSummary: According to a very recent paper [\textit{Z. Chen} and \textit{H. Li}, ``Energy quantization of the two dimensional Lane-Emden equation with vanishing potentials'',
Preprint, \url{arXiv:2310.05162}], the positive solution of the two-dimensional Hénon equation will concentrate on finite points when \(p \to \infty\). In this short note, we prove the uniform boundedness of the solution and then obtain a formula to specify the location of the concentration points by a Kirchoff-Routh type function.Necessary and sufficient condition for existence for a case of eigenvalues of multiplicity twohttps://zbmath.org/1540.352192024-09-13T18:40:28.020319Z"Korman, Philip"https://zbmath.org/authors/?q=ai:korman.philip-lSummary: We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.Existence and multiplicity of solutions for resonant-superlinear problemshttps://zbmath.org/1540.352202024-09-13T18:40:28.020319Z"Papageorgiou, Nikolaos S."https://zbmath.org/authors/?q=ai:papageorgiou.nikolaos-s"Vetro, Calogero"https://zbmath.org/authors/?q=ai:vetro.calogero"Vetro, Francesca"https://zbmath.org/authors/?q=ai:vetro.francescaThe authors study existence and multiplicity of nontrivial solutions for semilinear Dirichlet problems of the form
\begin{align*}
-\Delta u(z)=\hat{\lambda}_1u(z)+f(z,u^+(z))+\theta(z) \quad \text{in }\Omega, \quad u|_{\partial\Omega}=0,
\end{align*}
where \(\Omega\subseteq \mathbb{R}^N\) with \(N\geq 2\) is a bounded domain with a \(C^2\)-boundary \(\partial\Omega\), \(\hat{\lambda}_1>0\) is the principal eigenvalue of \((-\Delta,H^1_0(\Omega))\), \(u^+=\max\{u,0\}\), \(\theta \in L^\infty(\Omega)\) with \(\theta(z) \leq 0\) for a.a.\,\(z\in\Omega\) and \(f\colon \Omega \times \mathbb{R}\to\mathbb{R}\) is a Carathéodory function which exhibits superlinear growth. By applying variational tools from critical point theory together with truncation and comparison techniques as well as critical groups, the authors prove two multiplicity theorems producing two and three nontrivial solutions. In addition, it is shown that the problem cannot have negative solutions.
Reviewer: Patrick Winkert (Berlin)On the nodal set of solutions to some sublinear equations without homogeneityhttps://zbmath.org/1540.352212024-09-13T18:40:28.020319Z"Soave, Nicola"https://zbmath.org/authors/?q=ai:soave.nicola"Tortone, Giorgio"https://zbmath.org/authors/?q=ai:tortone.giorgioSummary: We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem
\[
-\Delta u = \lambda_+(u^+)^{p-1}-\lambda_-(u^-)^{q-1},
\]
where \(1 \leq p<q<2\), \(\lambda_+ > 0\), \(\lambda_- \geq 0\). The equation is characterized by the sublinear \textit{inhomogeneous} character of the right hand-side, which makes it difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.Remarks on the Harnack inequality for the elliptic \((p, q)\)-Laplacianhttps://zbmath.org/1540.352222024-09-13T18:40:28.020319Z"Aliyev, M. J."https://zbmath.org/authors/?q=ai:aliyev.mushviq-j"Alkhutov, Yu. A."https://zbmath.org/authors/?q=ai:alkhutov.yuriy-alexandrovich"Surnachev, M. D."https://zbmath.org/authors/?q=ai:surnachev.mikail-dmitrievich"Tikhomirov, R. N."https://zbmath.org/authors/?q=ai:tikhomirov.r-nSummary: We establish a new Harnack inequality for nonnegative solutions to the \(p(x)\)-Laplace equation with two-phase exponent \(p(x)\) taking two constant values \(p\) and \(q\) in the case where the phase interface is a hyperplane.Retraction notice to: ``Necessary optimality conditions of an optimization problem governed by a double phase PDE''https://zbmath.org/1540.352232024-09-13T18:40:28.020319Z"Benslimane, Omar"https://zbmath.org/authors/?q=ai:benslimane.omar"Gadhi, Nazih Abderrazzak"https://zbmath.org/authors/?q=ai:gadhi.nazih-abderrazzakSummary: The article [ibid. 524, No. 2, Article ID 127117, 16 p. (2023; Zbl 1512.35331)] has been retracted: please see Elsevier Policy on Article Withdrawal (\url{https://www.elsevier.com/about/policies/article-withdrawal}).
This article has been retracted at the request of the authors, Omar Benslimane and Nazih Abderrazzak Gadhi. Following publication, questions were raised by a reader regarding the mathematical derivations of Lemma 5, as well as in the formulation of Definition 1, in the paper, which the authors and the journal worked to resolve. In the process of corrections, the authors noticed inconsistencies between the investigated problem and the weak formulation of the equation that is associated with it. The above-named authors have concluded that such inconsistencies have rendered the main results in the paper erroneous. The authors apologise for these errors.Normalized solutions for \((p,q)\)-Laplacian equations with mass supercritical growthhttps://zbmath.org/1540.352242024-09-13T18:40:28.020319Z"Cai, Li"https://zbmath.org/authors/?q=ai:cai.li|cai.li.1"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-dThe authors study a \((p,q)\)-Laplace problem with \(L^p\)-constraint
\begin{align*}
\begin{cases} -\Delta_p u-\Delta_q u+\lambda|u|^{p-2}u=f(u) \quad \text{in }\mathbb{R}^N,\\
\int_{\mathbb{R}^N} |u|^p \,\mathrm{d}x=c^p,\\
u\in W^{1,p}(\mathbb{R^N}) \cap W^{1,q}(\mathbb{R}^N), \end{cases}
\end{align*}
where \(1<p<q<N\), \(\Delta_ru=\operatorname{div}(|\nabla u|^{r-2}\nabla u)\) for \(r\in\{p,q\}\) is the \(r\)-Laplacian, \(\lambda\) is a Lagrange multiplier, \(c>0\) and \(f\colon \mathbb{R}\to\mathbb{R}\) is a continuous function that satisfies weak mass supercritical assumptions. The authors prove the existence of ground states and analyze the behavior of the ground state energy \(E_c\) when \(c>0\) varies.
Reviewer: Patrick Winkert (Berlin)On Zaremba problem for \(p\)-elliptic equationhttps://zbmath.org/1540.352252024-09-13T18:40:28.020319Z"Chechkina, Aleksandra G."https://zbmath.org/authors/?q=ai:chechkina.aleksandra-grigorievnaSummary: Higher integrability for the gradient of the solution to the Zaremba problem in a bounded strictly Lipschitz domain for the inhomogeneous \(p\)-elliptic equation is proved.Calderón-Zygmund estimates for nonlinear equations of differential forms with BMO coefficientshttps://zbmath.org/1540.352262024-09-13T18:40:28.020319Z"Lee, Mikyoung"https://zbmath.org/authors/?q=ai:lee.mikyoung"Ok, Jihoon"https://zbmath.org/authors/?q=ai:ok.jihoon"Pyo, Juncheol"https://zbmath.org/authors/?q=ai:pyo.juncheolSummary: We obtain \(L^q\)-regularity estimates for weak solutions to \(p\)-Laplacian type equations of differential forms. In particular, we prove local Calderón-Zygmund type estimates for equations with discontinuous coefficients satisfying the bounded mean oscillation (BMO) condition.Monotonicity of solutions to degenerate \(p\)-Laplace problems with a gradient term in half-spaceshttps://zbmath.org/1540.352272024-09-13T18:40:28.020319Z"Le, Phuong"https://zbmath.org/authors/?q=ai:le.phuong-quynh|le.phuong-m"Huynh, Nhat Vy"https://zbmath.org/authors/?q=ai:huynh.nhat-vySummary: We establish the monotonicity of positive solutions to the problem
\[
-\Delta_p u + a(u)|\nabla u|^q = f(u) \text{ in } \mathbb{R}^N_+, \quad u=0 \text{ on } \partial \mathbb{R}^N_+,
\]
where \(p > 2\), \(q \geq p - 1\) and \(a\), \(f\) are locally Lipschitz continuous functions such that \(f\) is positive on \((0, +\infty)\) and it is either sublinear or superlinear near 0. The main tool we use is the refined method of moving planes for quasilinear elliptic problems in half-spaces.Gradient bounds for non-uniformly quasilinear elliptic two-sided obstacle problems with variable exponentshttps://zbmath.org/1540.352282024-09-13T18:40:28.020319Z"Minh-Phuong Tran"https://zbmath.org/authors/?q=ai:minh-phuong-tran."Thanh-Nhan Nguyen"https://zbmath.org/authors/?q=ai:thanh-nhan-nguyen."Le-Tuyet-Nhi Pham"https://zbmath.org/authors/?q=ai:le-tuyet-nhi-pham.Summary: We are concerned with a class of quasilinear elliptic variational inequalities, the so-called two-obstacle problems, that are driven by the double-phase operators with variable exponents. In this paper, by using the terminology of fractional maximal distribution functions, we prove decay estimates for level sets of the gradient of weak solutions in turn implying regularity estimates in some generalized weighted function spaces.Degenerated and competing anisotropic \((p,q)\)-Laplacians with weightshttps://zbmath.org/1540.352292024-09-13T18:40:28.020319Z"Razani, A."https://zbmath.org/authors/?q=ai:razani.abdolrahman"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcher(no abstract)A note on the classification of positive solutions to the critical \(p\)-Laplace equation in \(\mathbb{R}^n\)https://zbmath.org/1540.352302024-09-13T18:40:28.020319Z"Vétois, Jérôme"https://zbmath.org/authors/?q=ai:vetois.jeromeSummary: In this note, we obtain a classification result for positive solutions to the critical \(p\)-Laplace equation in \(\mathbb{R}^n\) with \(n\geq 4\) and \(p>p_n\) for some number \(p_n \in \left(\frac{n}{3},\frac{n+1}{3}\right)\) such that \(p_n \sim \frac{n}{3}+\frac{1}{n}\), which improves upon a similar result obtained by
\textit{Q. Ou} [``On the classification of entire solutions to the critical \(p\)-Laplace equation'', Preprint, \url{arXiv:2210.05141}] under the condition \(p\geq \frac{n+1}{3}\).On a nonlinear equation \(p(x)\)-elliptic problem of Neumann type by topological degree methodhttps://zbmath.org/1540.352312024-09-13T18:40:28.020319Z"Yacini, Soukaina"https://zbmath.org/authors/?q=ai:yacini.soukaina"Abbassi, Adil"https://zbmath.org/authors/?q=ai:abbassi.adil"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakir"Kassidi, Abderrazak"https://zbmath.org/authors/?q=ai:kassidi.abderrazakSummary: The aim of this work is devoted to study the existence of weak solutions for the nonlinear \(p(x)\)-elliptic problem,
\[
-\mathrm{div}\,a(x,u,\nabla u)=b(x)|u|^{p(x)-2}u+\lambda H(x,u,\nabla u) \quad \text{in} \quad \varOmega,
\]
in the Weighted Sobolev spaces Weighted-withe Exponent Variable. The existence is proved by using the topological degree, introduced by Berkovits.
For the entire collection see [Zbl 1515.35012].Existence and multiplicity of nontrivial solutions for a \((p,q)\)-Laplacian system on locally finite graphshttps://zbmath.org/1540.352322024-09-13T18:40:28.020319Z"Yang, Ping"https://zbmath.org/authors/?q=ai:yang.ping"Zhang, Xingyong"https://zbmath.org/authors/?q=ai:zhang.xingyongSummary: We generalize two embedding theorems and investigate the existence and multiplicity of nontrivial solutions for a \((p,q)\)-Laplacian coupled system with perturbations and two parameters \(\lambda_1\) and \(\lambda_2\) on locally finite graph. By using the Ekeland's variational principle, we obtain that system has at least one nontrivial solution when the nonlinear term satisfies the sub-\((p,q)\) conditions. We also obtain a necessary condition for the existence of semi-trivial solutions to the system. Moreover, by using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one solution of positive energy and one solution of negative energy when the nonlinear term satisfies the super-\((p,q)\) conditions which is weaker than the well-known Ambrosetti-Rabinowitz condition. Especially, in all of the results, we present the concrete ranges of the parameters \(\lambda_1\) and \(\lambda_2\).Hölder continuity of the gradients for non-homogenous elliptic equations of \(p(x)\)-Laplacian typehttps://zbmath.org/1540.352332024-09-13T18:40:28.020319Z"Yao, Fengping"https://zbmath.org/authors/?q=ai:yao.fengpingSummary: The main goal of this paper is to discuss the local Hölder continuity of the gradients for weak solutions of the following non-homogenous elliptic \(p(x)\)-Laplacian equations of divergence form
\[
\begin{aligned} \operatorname{div} \left(\left( A(x) \nabla u(x) \cdot \nabla u(x) \right)^{\frac{p(x)-2}{2}} A(x) \nabla u(x) \right) = \operatorname{div} \left(|\mathbf{f}(x) |^{p(x)-2} \mathbf{f}(x) \right) \text{ in } \Omega, \end{aligned}
\]
where \(\Omega \subset \mathbb{R}^n\) is an open bounded domain for \(n \geq 2\), under some proper non-Hölder conditions on the variable exponents \(p(x)\) and the coefficients matrix \(A(x)\).Regularity for the Monge-Ampère equation by doublinghttps://zbmath.org/1540.352342024-09-13T18:40:28.020319Z"Shankar, Ravi"https://zbmath.org/authors/?q=ai:shankar.ravi-v|shankar.ravi|shankar.ravi.1|shankar.ravi.2"Yuan, Yu"https://zbmath.org/authors/?q=ai:yuan.yu.3|yuan.yuSummary: We give a new proof for the interior regularity of strictly convex solutions of the Monge-Ampère equation. Our approach uses a doubling inequality for the Hessian in terms of the extrinsic distance function on the maximal Lagrangian submanifold determined by the potential equation.General kernel estimates of Schrödinger-type operators with unbounded diffusion termshttps://zbmath.org/1540.352362024-09-13T18:40:28.020319Z"Caso, Loredana"https://zbmath.org/authors/?q=ai:caso.loredana"Kunze, Markus"https://zbmath.org/authors/?q=ai:kunze.markus-christian"Porfido, Marianna"https://zbmath.org/authors/?q=ai:porfido.marianna.1"Rhandi, Abdelaziz"https://zbmath.org/authors/?q=ai:rhandi.abdelazizSummary: We first prove that the realization \(A_{\min}\) of \(A:=\operatorname{div}(Q\nabla) -V\) in \(L^2 (\mathbb{R}^d)\) with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on \(L^2 (\mathbb{R}^d)\) which coincides on \(L^2 (\mathbb{R}^d)\cap C_b (\mathbb{R}^d)\) with the minimal semigroup generated by a realization of \(A\) on \(C_b (\mathbb{R}^d)\). Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of \(A\) and deduce some spectral properties of \(A_{\min}\) in the case of polynomially and exponentially growing diffusion and potential coefficients.Heat coefficients for magnetic Laplacians on the complex projective space \(\mathbb{P}(\mathbb{C})\)https://zbmath.org/1540.352672024-09-13T18:40:28.020319Z"Ahbli, K."https://zbmath.org/authors/?q=ai:ahbli.khalid"Hafoud, A."https://zbmath.org/authors/?q=ai:hafoud.ali"Mouayn, Z."https://zbmath.org/authors/?q=ai:mouayn.zouhairSummary: We denote by \(\Delta_\nu\) the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to \(\nu\). When acting on bounded functions on the complex projective \(n\)-space, this operator has a discrete spectrum consisting on eigenvalues \(\beta_m\), \(m\in \mathbb{Z}_+\). For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of \(\Delta_\nu\). Using a suitable polynomial decomposition of the multiplicity of each \(\beta_m\), we write down a trace formula for the heat operator associated with \(\Delta_\nu\) in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as \(t\searrow 0^+\) by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with \(\Delta_\nu\).On shifting the principal eigenvalue of Dirichlet problem to infinity with non-transversal incompressible drifthttps://zbmath.org/1540.352682024-09-13T18:40:28.020319Z"Athmouni, Nassim"https://zbmath.org/authors/?q=ai:athmouni.nassim"Damak, Mondher"https://zbmath.org/authors/?q=ai:damak.mondher"Franke, Brice"https://zbmath.org/authors/?q=ai:franke.brice"Yaakoubi, Nejib"https://zbmath.org/authors/?q=ai:yaakoubi.nejibSummary: We prove that it is always possible to add some divergence free drift vector field to some two dimensional spherical Dirichlet problem, such that the resulting principal eigenvalue lies above a prescribed bound. By construction those drift vector fields vanish on the boundary and their flow lines individually stay away from the boundary. The capacity of those drift vector fields to accelerate diffusivity originates from high frequency oscillation of the associated flow lines. The lower bounds for the spectrum are obtained through isoperimetric inequalities for flow invariant functions.On Pleijel's nodal domain theorem for the Robin problemhttps://zbmath.org/1540.352692024-09-13T18:40:28.020319Z"Hassannezhad, Asma"https://zbmath.org/authors/?q=ai:hassannezhad.asma"Sher, David"https://zbmath.org/authors/?q=ai:sher.david-a|sher.david-bSummary: We prove an improved Pleijel nodal domain theorem for the Robin eigenvalue problem. In particular, we remove the restriction, imposed in previous work, that the Robin parameter be non-negative. We also improve the upper bound in the statement of the Pleijel theorem. In the particular example of a Euclidean ball, we calculate the explicit value of the Pleijel constant for a generic constant Robin parameter, and we show that it is equal to the Pleijel constant for the Dirichlet Laplacian on a Euclidean ball.
{\copyright} 2024 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.Estimates for the lowest Neumann eigenvalues of parallelograms and domains of constant widthhttps://zbmath.org/1540.352702024-09-13T18:40:28.020319Z"Léna, Corentin"https://zbmath.org/authors/?q=ai:lena.corentin"Rohleder, Jonathan"https://zbmath.org/authors/?q=ai:rohleder.jonathanSummary: We prove sharp upper bounds for the first and second non-trivial eigenvalues of the Neumann Laplacian in two classes of domains: parallelograms and domains of constant width. This gives in particular a new proof of an isoperimetric inequality for parallelograms recently obtained by A. Henrot, A. Lemenant and I. Lucardesi.Asymmetric transport for magnetic Dirac equationshttps://zbmath.org/1540.352712024-09-13T18:40:28.020319Z"Quinn, Solomon"https://zbmath.org/authors/?q=ai:quinn.solomon"Bal, Guillaume"https://zbmath.org/authors/?q=ai:bal.guillaumeSummary: This paper concerns the asymmetric transport associated with a low-energy interface Dirac model of graphene-type materials subject to external magnetic and electric fields. We show that the relevant physical observable, an interface conductivity, is quantized and robust to a large class of perturbations. These include defects that decay along or away from the interface, and sufficiently small or localized changes in the external fields. An explicit formula for the interface conductivity is given by a spectral flow.Quantum tunneling in deep potential wells and strong magnetic field revisitedhttps://zbmath.org/1540.352732024-09-13T18:40:28.020319Z"Helffer, Bernard"https://zbmath.org/authors/?q=ai:helffer.bernard"Kachmar, Ayman"https://zbmath.org/authors/?q=ai:kachmar.aymanSummary: We investigate a Hamiltonian with a symmetric double well and a uniform magnetic field, where tunneling occurs in the simultaneous limit of strong magnetic field and deep potential wells with disjoint supports. We derive an accurate estimate of its magnitude, and obtain a precise leading order asymptotic expression for the effect of a strong magnetic field, improving on the upper and lower bounds established earlier by Fefferman, Shapiro and Weinstein.Gaps in the spectrum of thin waveguides with periodically locally deformed wallshttps://zbmath.org/1540.352742024-09-13T18:40:28.020319Z"Nazarov, S. A."https://zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovichSummary: The structures of quantum and acoustic waveguides obtained by joining a periodic family of small knots to a thin cylinder are examined. Asymptotic expansions of eigenvalues of a model problem in the periodicity cell are obtained, which are used to derive asymptotic formulas for the disposition and sizes of the gaps in the spectra of the corresponding Dirichlet and Neumann problems for the Laplace operator. Geometric and integral characteristics of the waveguides are found that ensure the opening of several spectral gaps.The scattering resonances for Schrödinger-type operators with unbounded potentialshttps://zbmath.org/1540.352762024-09-13T18:40:28.020319Z"Li, Peijun"https://zbmath.org/authors/?q=ai:li.peijun.1"Yao, Xiaohua"https://zbmath.org/authors/?q=ai:yao.xiaohua"Zhao, Yue"https://zbmath.org/authors/?q=ai:zhao.yue.1In this paper, the authors study the meromorphic continuation of the outgoing resolvent associated with Schrödinger-type operators in three dimensions. This area closely relates to the theory of scattering resonances, which are defined as poles in the meromorphic continuation. And scattering resonances phenomena have significant applications across various scientific and engineering research fields. The existing research on the meromorphic continuation of resolvents primarily concentrates on bounded potentials. This work extends some of these results to accommodate unbounded potentials for Schrödinger-type operators, including the classical Schrödinger-type operator involving unbounded potentials and the fractional Schrödinger operator with both bounded and unbounded potentials. The analysis relies on a resolvent identity that establishes a connection between the resolvents of the fractional Schrödinger operator and its classical counterpart.
Reviewer: Jiqiang Zheng (Beijing)Asymptotic behavior and monotonicity of radial eigenvalues for the \(p\)-Laplacianhttps://zbmath.org/1540.352772024-09-13T18:40:28.020319Z"Kajikiya, Ryuji"https://zbmath.org/authors/?q=ai:kajikiya.ryuji"Tanaka, Mieko"https://zbmath.org/authors/?q=ai:tanaka.mieko"Tanaka, Satoshi"https://zbmath.org/authors/?q=ai:tanaka.satoshiIn this paper ,the authors are concerned with the study of some properties of the radial eigenvalues of the \(p\)-Laplace operator, under the homogeneous Dirichlet boundary conditions, on either a ball or an annulus. More precisely, for each positive integer \(k\) and each real number \(p\in(1,\infty)\), letting \(\lambda_k(p)\) to be the \(k\)-th radial eigenvalue of the \(p\)-Laplace operator are investigated, on the one hand, the asymptotic behavior of \(\lambda_k(p)\) as \(p\rightarrow 1^+\) or as \(p\rightarrow\infty\) and, on the other hand, the monotonicity (or non-monotonicity) of the function \((1,\infty)\ni p\mapsto\lambda_k(p)\). In particular, the results from this paper complement earlier studies from the topic.
Reviewer: Mihai Mihăilescu (Craiova)Structural stability and optimal convergence rates of subsonic Euler flows with large vorticity in infinitely long nozzleshttps://zbmath.org/1540.352992024-09-13T18:40:28.020319Z"Liao, Jing"https://zbmath.org/authors/?q=ai:liao.jing"Tan, Zhong"https://zbmath.org/authors/?q=ai:tan.zhong.1|tan.zhongSummary: In this paper, we establish the structural stability and optimal convergence rates of non-isentropic subsonic Euler flows with large vorticity in two-dimensional infinitely long nozzles. By applying the stream function formulation for compressible Euler equations, Euler equations are equivalent to a quasilinear second order elliptic equation of the stream function for subsonic flow. Then, the key points to prove the structural stability of subsonic flows with large vorticity under small perturbations of nozzle boundaries are the standard estimates of elliptic equations. Furthermore, using the maximum principle and the choice of compared functions, we obtain the optimal convergence rates of subsonic flows at far fields.Traveling wave phenomena of inhomogeneous half-wave equationhttps://zbmath.org/1540.353702024-09-13T18:40:28.020319Z"Feng, Zhaosheng"https://zbmath.org/authors/?q=ai:feng.zhaosheng"Su, Yu"https://zbmath.org/authors/?q=ai:su.yu.4|su.yu|su.yu.3|su.yu.2|su.yu.1Summary: In this paper, we are concerned with traveling wave phenomena of the inhomogeneous half-wave equation, which models the energy of a spin zero particle in the Coulomb field. We study the Gagliardo-Nirenberg and critical Hardy-Sobolev inequalities with velocity \(0 < |v| < 1\) and obtain the estimates for the best constants and optimizers of inequalities. Moreover, we establish the non-scattering results with small traveling wave for energy subcritical and critical cases.Groundstates for planar Schrödinger-Poisson system involving convolution nonlinearity and critical exponential growthhttps://zbmath.org/1540.353752024-09-13T18:40:28.020319Z"Jin, Peng"https://zbmath.org/authors/?q=ai:jin.peng"Shu, Muhua"https://zbmath.org/authors/?q=ai:shu.muhua"Wen, Lixi"https://zbmath.org/authors/?q=ai:wen.lixiSummary: This paper is concerned with a planar Schrödinger-Poisson system involving Stein-Weiss nonlinearity
\[
\begin{cases}
-\Delta u+V(x)u+\phi u=\frac{1}{|x|^{\beta}} \left( \int_{\mathbb{R}^2}\frac{F(u(y))}{|x-y|^{\mu} |y|^{\beta}}dy\right) f(u), & x\in \mathbb{R}^2, \\
\Delta \phi =u^2, & x\in \mathbb{R}^2,
\end{cases}
\tag{0.1}
\]
and its degenerate case
\[
\begin{cases}
-\Delta u+\phi u= \left( \int_{\mathbb{R}^2}\frac{F(u(y))}{|x-y|^{\mu}}dy\right) f(u), & x\in \mathbb{R}^2, \\
\Delta \phi =u^2, & x\in \mathbb{R}^2,
\end{cases}
\tag{0.2}
\]
where \(\beta \geq 0\), \(0<\mu <2\), \(2\beta +\mu <2\), \(V\in \mathcal{C}(\mathbb{R}^2,\mathbb{R})\) and \(f\) is of exponential critical growth. By combining variational methods, Stein-Weiss inequality and some delicate analysis, we derive the existence of ground state solution for the first system. Under some mild assumptions, we introduce the Pohozaev identity of the equivalent equation of the second system and use Jeanjean's monotonicity method to achieve the existence of nontrivial solution for the second system.Threshold for existence, non-existence and multiplicity of positive solutions with prescribed mass for an NLS with a pure power nonlinearity in the exterior of a ballhttps://zbmath.org/1540.353802024-09-13T18:40:28.020319Z"Song, Linjie"https://zbmath.org/authors/?q=ai:song.linjie"Hajaiej, Hichem"https://zbmath.org/authors/?q=ai:hajaiej.hichemSummary: We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem for the \(L^2\) supercritical case. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent \(p = 6\) when \(N = 2\), which does not seem to have played a role for this equation in the past. Moreover, our findings are ``quite surprising'' and completely different from the results obtained on the entire space and on balls. We will also show that the nature of the domain is crucial for the existence and stability of standing waves. As a foretaste, it is well-known that in the supercritical case these waves are unstable in \(\mathbb{R}^N\). In this paper, we will show that in the exterior domain they are strongly stable.Sharp criterion of global existence for a class of nonlinear Schrödinger equations with critical exponenthttps://zbmath.org/1540.353822024-09-13T18:40:28.020319Z"Xu, Runzhang"https://zbmath.org/authors/?q=ai:xu.runzhang"Wang, Xuemei"https://zbmath.org/authors/?q=ai:wang.xuemei"Niu, Yi"https://zbmath.org/authors/?q=ai:niu.yi"Zhang, Mingyou"https://zbmath.org/authors/?q=ai:zhang.mingyou"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie.21Summary: This paper discusses the inhomogeneous nonlinear Schrödinger equation with critical exponent. By constructing a variational problem and the so-called invariant manifolds of the evolution flow, we derive a sharp criterion for blowup and global existence of the solutions.Local minimality of \(\mathbb{R}^N\)-valued and \(\mathbb{S}^N\)-valued Ginzburg-Landau vortex solutions in the unit ball \(B^N\)https://zbmath.org/1540.353832024-09-13T18:40:28.020319Z"Ignat, Radu"https://zbmath.org/authors/?q=ai:ignat.radu"Nguyen, Luc"https://zbmath.org/authors/?q=ai:nguyen.lucSummary: We study the existence, uniqueness and minimality of critical points of the form \(m_{\varepsilon,\eta}(x)=(f_{\varepsilon,\eta}(|x|)\frac{x}{|x|},g_{\varepsilon,\eta}(|x|))\) of the functional \(E_{\varepsilon,\eta}[m]=\int_{B^N}[\frac{1}{2}|\nabla m|^2+\frac{1}{4\varepsilon^2}(1-|m|^2)^2+\frac{1}{2\eta^2}m^2_{N+1}]dx\) for \(m=(m_1,\dots,m_N,M_{N+1})\in H^1(B^N,\mathbb{R}^{N+1})\) with \(m(x)=(x,0)\) on \(\partial B^N\). We establish a necessary and sufficient condition on the dimension \(N\) and the parameters \(\varepsilon\) and \(\eta\) for the existence of an escaping vortex solution \((f_{\varepsilon,\eta},g_{\varepsilon,\eta})\) with \(g_{\varepsilon,\eta}>0\). We also establish its uniqueness and local minimality. In particular, when \(\eta=0\), we prove the local minimality of the degree-one vortex solution for the Ginzburg-Landau (GL) energy for every \(\varepsilon>0\) and \(N\geq 2\). Similarly, when \(\varepsilon=0\), we prove the local minimality of the degree-one escaping vortex solution to an \(\mathbb{S}^N\)-valued GL model in micromagnetics for all \(\eta>0\) and \(2\leq N\leq 6\).Compact embeddings, eigenvalue problems, and subelliptic Brezis-Nirenberg equations involving singularity on stratified Lie groupshttps://zbmath.org/1540.354262024-09-13T18:40:28.020319Z"Ghosh, Sekhar"https://zbmath.org/authors/?q=ai:ghosh.sekhar"Kumar, Vishvesh"https://zbmath.org/authors/?q=ai:kumar.vishvesh"Ruzhansky, Michael"https://zbmath.org/authors/?q=ai:ruzhansky.michael-vThe authors consider the eigenvalue problem associated with the fractional \(p\)-sub-Laplacian on a stratified Lie group \(\mathbb{G}\) and investigate several properties of the corresponding first eigenpair: the existence of the first eigenfunction and its boundedness, a lower estimate on \(\lambda_1\), simplicity of \(\lambda_1\) and its isolation in the spectrum. Using this information and methods of the Nehari manifold, the authors establish the existence of at least two solutions to a parametric family of nonlocal singular subelliptic equations, and investigate their regularity. As an important auxiliary result, the compactness of the embedding of a space \(X_0^{s,p}(\Omega)\) into \(L^r(\Omega)\) is obtained, where \(X_0^{s,p}(\Omega)\) is defined as the closure of \(C_0^\infty(\Omega)\) in \(W^{s,p}(\mathbb{G})\).
Reviewer: Vladimir Bobkov (Ufa)Asymptotically linear magnetic fractional problemshttps://zbmath.org/1540.354332024-09-13T18:40:28.020319Z"Bartolo, Rossella"https://zbmath.org/authors/?q=ai:bartolo.rossella"d'Avenia, Pietro"https://zbmath.org/authors/?q=ai:davenia.pietro"Molica Bisci, Giovanni"https://zbmath.org/authors/?q=ai:molica-bisci.giovanniSummary: The aim of this paper is investigating the existence and multiplicity of weak solutions to non-local equations involving the \textit{magnetic fractional Laplacian}, when the nonlinearity is subcritical and asymptotically linear at infinity. We prove existence and multiplicity results by using variational tools, extending to the magnetic local and non-local setting some known results for the classical and the fractional Laplace operators.Boundary estimates and a Wiener criterion for the fractional Laplacianhttps://zbmath.org/1540.354352024-09-13T18:40:28.020319Z"Björn, Jana"https://zbmath.org/authors/?q=ai:bjorn.janaIn this paper, the author used the Caffarelli-Silvestre extension to show for a general open set \(\Omega\subset \mathbb{R}^n\) that a boundary point \(x_0\) is regular for the fractional Laplace equation \((-\Delta)^su=0\), \(0<s<1\), if and only if \((x_0, 0)\) is regular for the extended weighted equation in a subset of \(\mathbb{R}^{n+1}\). By using a Wiener criterion and a Besov capacity, a Mazya-type boundary estimate was established.
Reviewer: Mingqi Xiang (Tianjin)Nonexistence of anti-symmetric solutions for fractional Hardy-Hénon systemhttps://zbmath.org/1540.354432024-09-13T18:40:28.020319Z"Hu, Jiaqi"https://zbmath.org/authors/?q=ai:hu.jiaqi"Du, Zhuoran"https://zbmath.org/authors/?q=ai:du.zhuoranSummary: We study anti-symmetric solutions about the hyperplane \(\{x_n =0\}\) for the following fractional Hardy-Hénon system:
\[
\begin{cases}
(-\Delta)^{s_1}u(x)=|x|^{\alpha} v^p (x), & x\in\mathbb{R}_+^n, \\
(-\Delta)^{s_2}v(x)=|x|^{\beta} u^q (x), & x\in\mathbb{R}_+^n, \\
u(x)\geq 0, & v(x)\geqslant 0, x\in\mathbb{R}_+^n,
\end{cases}
\]
where \(0< s_1, s_2 <1, n>2\max \{s_1,s_2\}\). Nonexistence of anti-symmetric solutions are obtained in some appropriate domains of \((p,q)\) under some corresponding assumptions of \(\alpha,\beta\) via the methods of moving spheres and moving planes.
Particularly, for the case \(s_1 =s_2\), one of our results shows that one domain of \((p,q)\), where nonexistence of anti-symmetric solutions with appropriate decay conditions at infinity hold true, locates at above the fractional Sobolev's hyperbola under appropriate condition of \(\alpha,\beta\).Existence and multiplicity of non-trivial solutions for fractional Schrödinger-Poisson systems with a combined nonlinearityhttps://zbmath.org/1540.354502024-09-13T18:40:28.020319Z"Soluki, M."https://zbmath.org/authors/?q=ai:soluki.m"Afrouzi, G. A."https://zbmath.org/authors/?q=ai:afrouzi.ghasem-alizadeh"Rasouli, S. H."https://zbmath.org/authors/?q=ai:rasouli.sayyed-hashemSummary: In this paper, we are concerned with the following fractional Schrödinger-Poisson system:
\[
\begin{cases}
(-\Delta)^s u + V(x)u +\lambda \phi u=K(x) |u|^{q-2} u+ f(x, u), & \qquad x \in \mathbb{R}^3 \\
(-\Delta)^t \phi = u^2, & \qquad x \in \mathbb{R}^3
\end{cases}
\]
where \(\lambda > 0\) is a constant, \(s,t \in (0,1]\), \(2t+4s>3\), \(1<q<2\) and \(f(x, u)\) is linearly bounded in \(u\) at infinity. With some assumptions on \(K, V\) and \(f\) we get the existence and multiplicity of non-trivial solutions with the help of the variational methods.Eigenvalue type problem in \(s(.,.)\)-fractional Musielak-Sobolev spaceshttps://zbmath.org/1540.354512024-09-13T18:40:28.020319Z"Srati, Mohammed"https://zbmath.org/authors/?q=ai:srati.mohammedSummary: In this paper, we introduce the \(s(., .)\)-fractional Musielak-Sobolev spaces \(W^{s(x,y)}L_{\varPhi_{x,y}}(\Omega)\). Then, we show that there exists \(\lambda_*>0\) such that any \(\lambda \in (0, \lambda_*)\) is an eigenvalue for the following problem, by means of Ekeland's variational principle
\[
(\mathcal{P}_a)
\begin{cases}
(-\Delta)^{s(x,.)}_{a_{(x,.)}} u = \lambda |u|^{q(x)-2}u & \text{ in } \Omega, \\
\qquad \qquad \; u = 0 & \text{ in } \mathbb{R}^N \setminus \Omega,
\end{cases}
\]
where \(\Omega\) is a bounded open subset of \(\mathbb{R}^N\) with \(C^{0,1}\)-regularity and bounded boundary.Determining surface heat flux for noncharacteristic Cauchy problem for Laplace equationhttps://zbmath.org/1540.354602024-09-13T18:40:28.020319Z"Zhao, Jing-jun"https://zbmath.org/authors/?q=ai:zhao.jingjun"Liu, Song-shu"https://zbmath.org/authors/?q=ai:liu.songshu"Liu, Tao"https://zbmath.org/authors/?q=ai:liu.tao.1Summary: In this paper, the noncharacteristic Cauchy problem for the Laplace equation
\[
\begin{cases}
w_{x x} + w_{y y} = 0 \quad & x \in(0, 1), \quad y \in R, \\
w(0, y) = g(y) \quad & y \in R, \\
w_x(0, y) = h(y) \quad & y \in R,
\end{cases}
\]
is investigated, where the Cauchy data is given at \(x = 0\) and the heat flux is sought in the interval \(0 < x \leq 1\). This problem is severely ill-posed: the solution (if it exists) does not depend continuously on the given data. A modified regularization method is used to solve this problem. Furthermore, some error estimates for the heat flux between the regularization solution and the exact solution are given. Finally, a numerical example shows that the proposed method works well.The linearized partial data Calderón problem for biharmonic operatorshttps://zbmath.org/1540.354612024-09-13T18:40:28.020319Z"Agrawal, Divyansh"https://zbmath.org/authors/?q=ai:agrawal.divyansh"Jaiswal, Ravi Shankar"https://zbmath.org/authors/?q=ai:jaiswal.ravi-shankar"Sahoo, Suman Kumar"https://zbmath.org/authors/?q=ai:sahoo.suman-kumarSummary: We consider a linearized partial data Calderón problem for biharmonic operators extending the analogous result for harmonic operators [\textit{D. Dos Santos Ferreira} et al., Math. Res. Lett. 16, No. 5--6, 955--970 (2009; Zbl 1198.31003)]. We construct special solutions and utilize Segal-Bargmann transform to recover lower order perturbations.Early-warning inverse source problem for the elasto-gravitational equationshttps://zbmath.org/1540.354622024-09-13T18:40:28.020319Z"Baldassari, L."https://zbmath.org/authors/?q=ai:baldassari.lorenzo"de Hoop, M. V."https://zbmath.org/authors/?q=ai:de-hoop.maarten-v"Francini, E."https://zbmath.org/authors/?q=ai:francini.elisa"Vessella, S."https://zbmath.org/authors/?q=ai:vessella.sergioSummary: Through coupled physics, we study an early-warning inverse source problem for the constant-coefficient elasto-gravitational equations. It consists of a mixed hyperbolic-elliptic system of partial differential equations describing elastic wave displacement and gravity perturbations produced by a source in a homogeneous bounded medium. Within the Cowling approximation, we prove uniqueness and Lipschitz stability for the inverse problem of recovering the moment tensor and the location of the source from early-time measurements of the changes of the gravitational field. The setup studied in this paper is motivated by gravity-based earthquake early warning systems, which are gaining much attention recently.Some computational tests for inverse conductivity problems based on vector, variational principles: the 2D casehttps://zbmath.org/1540.354632024-09-13T18:40:28.020319Z"Bandeira, L."https://zbmath.org/authors/?q=ai:bandeira.luis"Pedregal, P."https://zbmath.org/authors/?q=ai:pedregal.pablo(no abstract)Reconstruction of cracks in Calderón's inverse conductivity problem using energy comparisonshttps://zbmath.org/1540.354662024-09-13T18:40:28.020319Z"Garde, Henrik"https://zbmath.org/authors/?q=ai:garde.henrik"Vogelius, Michael"https://zbmath.org/authors/?q=ai:vogelius.michael-sLet \(\Omega \subset \mathbb{R}^d\) be a bounded Lipschitz domain with connected complement. A conductivity equation of the form \(- \nabla \cdot (\gamma_0 \nabla u) = 0\) is considered in \(\Omega \setminus D\), where \(D\) is the union of some Lipschitz hypersurfaces, called ``cracks''. Moreover, interface conditions on the cracks and homogeneous Neumann boundary conditions a part of the boundary of \(\Omega\) are imposed. The inverse problem considered consists in recovering the location of the cracks from the knowledge of a local Neumann-to-Dirichlet map on the remainder of the boundary. Reconstruction of the cracks is achieved by means of two operator monotonicity results for the local Neumann-to-Dirichlet map.
Reviewer: Jonathan Rohleder (Stockholm)Partial data inverse problems for nonlinear magnetic Schrödinger equationshttps://zbmath.org/1540.354702024-09-13T18:40:28.020319Z"Lai, Ru-Yu"https://zbmath.org/authors/?q=ai:lai.ru-yu"Zhou, Ting"https://zbmath.org/authors/?q=ai:zhou.tingSummary: We prove that the knowledge of the Dirichlet-to-Neumann map, measured on a part of the boundary of a bounded domain in \(\mathbb{R}^n\), \(n\geq 2\), can uniquely determine, in a nonlinear magnetic Schrödinger equation, the vector-valued magnetic potential and the scalar electric potential, both being nonlinear in the solution.Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth caseshttps://zbmath.org/1540.354852024-09-13T18:40:28.020319Z"Grubb, Gerd"https://zbmath.org/authors/?q=ai:grubb.gerdSummary: Let \(P\) be a symmetric \(2a\)-order classical strongly elliptic pseudodifferential operator with \textit{even} symbol \(p(x,\xi)\) on \(\mathbb{R}^n\) \((0<a<1)\), for example a perturbation of \((-\Delta)^a\). Let \(\Omega\subset\mathbb{R}^n\) be bounded, and let \(P_D\) be the Dirichlet realization in \(L_2(\Omega)\) defined under the exterior condition \(u=0\) in \(\mathbb{R}^n\setminus\Omega\). When \(p(x,\xi)\) and \(\Omega\) are \(C^\infty\), it is known that the eigenvalues \(\lambda_j\) (ordered in a nondecreasing sequence for \(j\to \infty)\) satisfy a Weyl asymptotic formula
\[
\lambda_j(P_D)=C(P,\Omega)j^{2a/n}+o(j^{2a/n})\text{ for }j\to\infty,
\]
with \(C(P,\Omega)\) determined from the principal symbol of \(P\). We now show that this result is valid for more general operators with a possibly nonsmooth \(x\)-dependence, over Lipschitz domains, and that it extends to \(\tilde{P}=P+P'+P''\), where \(P''\) is an operator of order \(<\min\{2a, a+\frac 12\}\) with certain mapping properties, and \(P''\) is bounded in \(L_2(\Omega)\) (e.g. \(P''=V(x)\in L_\infty(\Omega))\). Also the regularity of eigenfunctions of \(P_D\) is discussed.Boundary value problems and Heisenberg uniqueness pairshttps://zbmath.org/1540.420132024-09-13T18:40:28.020319Z"Rigat, S."https://zbmath.org/authors/?q=ai:rigat.stephane"Wielonsky, F."https://zbmath.org/authors/?q=ai:wielonsky.franckSummary: We describe a general method for constructing Heisenberg uniqueness pairs \((\Gamma, \Lambda)\) in the euclidean space \(\mathbb{R}^n\) based on the study of boundary value problems for partial differential equations. As a result, we show, for instance, that any pair made of the boundary \(\Gamma\) of a bounded convex set \(\Omega\) and a sphere \(\Lambda\) is a Heisenberg uniqueness pair if and only if the square of the radius of \(\Lambda\) is not an eigenvalue of the Laplacian on \(\Omega\). The main ingredients for the proofs are the Paley-Wiener theorem, the uniqueness of a solution to a homogeneous Dirichlet or initial boundary value problem, the continuity of single layer potentials, and some complex analysis in \(\mathbb{C}^n\). Denjoy's theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful.Estimate for the intrinsic square function on \(p\)-adic Herz spaces with variable exponenthttps://zbmath.org/1540.420432024-09-13T18:40:28.020319Z"Sultan, Mehvish"https://zbmath.org/authors/?q=ai:sultan.mehvish"Sultan, Babar"https://zbmath.org/authors/?q=ai:sultan.babarSummary: Our aim in this paper is to define \(p\)-adic Herz spaces with variable exponents and prove the boundedeness of \(p\)-adic intrinsic square function in these spaces.Boundary value problems in Euclidean space for bosonic Laplacianshttps://zbmath.org/1540.420452024-09-13T18:40:28.020319Z"Ding, Chao"https://zbmath.org/authors/?q=ai:ding.chao"Phuoc-Tai Nguyen"https://zbmath.org/authors/?q=ai:phuoc-tai-nguyen."Ryan, John"https://zbmath.org/authors/?q=ai:ryan.johnSummary: A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group, in this case, the irreducible representation spaces of homogeneous harmonic polynomials. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with \(L^p\) boundary data, \(1 \le p \le \infty\). We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc.Sobolev orthogonal polynomials and spectral methods in boundary value problemshttps://zbmath.org/1540.420472024-09-13T18:40:28.020319Z"Fernández, Lidia"https://zbmath.org/authors/?q=ai:fernandez.lidia"Marcellán, Francisco"https://zbmath.org/authors/?q=ai:marcellan-espanol.francisco"Pérez, Teresa E."https://zbmath.org/authors/?q=ai:perez.teresa-e"Piñar, Miguel A."https://zbmath.org/authors/?q=ai:pinar.miguel-aSummary: In the variational formulation of a boundary value problem for the harmonic oscillator, Sobolev inner products appear in a natural way. First, we study the sequences of Sobolev orthogonal polynomials with respect to such an inner product. Second, their representations in terms of a sequence of Gegenbauer polynomials are deduced as well as an algorithm to generate them in a recursive way is stated. The outer relative asymptotics between the Sobolev orthogonal polynomials and classical Legendre polynomials is obtained. Next we analyze the solution of the boundary value problem in terms of a Fourier-Sobolev projector. Finally, we provide numerical tests concerning the reliability and accuracy of the Sobolev spectral method.A characterization of the individual maximum and anti-maximum principlehttps://zbmath.org/1540.470542024-09-13T18:40:28.020319Z"Arora, Sahiba"https://zbmath.org/authors/?q=ai:arora.sahiba"Glück, Jochen"https://zbmath.org/authors/?q=ai:gluck.jochenSummary: Abstract approaches to maximum and anti-maximum principles for differential operators typically rely on the condition that all vectors in the domain of the operator are dominated by the leading eigenfunction of the operator. We study the necessity of this condition. In particular, we show that under a number of natural assumptions, so-called individual versions of both the maximum and the anti-maximum principle simultaneously hold if and only if the aforementioned domination condition is satisfied. Consequently, we are able to show that a variety of concrete differential operators do not satisfy an anti-maximum principle.Strong uniqueness of finite-dimensional Dirichlet operators with singular driftshttps://zbmath.org/1540.470662024-09-13T18:40:28.020319Z"Lee, Haesung"https://zbmath.org/authors/?q=ai:lee.haesungSummary: We show \(L^r (\mathbb{R}^d, \mu)\)-uniqueness for any \(r\in (1,2]\) and the essential self-adjointness of a Dirichlet operator \(Lf=\Delta f+\langle\frac{1}{\rho}\nabla\rho,\nabla f\rangle\), \(f\in C_0^{\infty}(\mathbb{R}^d)\) with \(d\geq 3\) and \(\mu=\rho dx\). In particular, \(\nabla\rho\) is allowed to be in \(L_{\mathrm{loc}}^d (\mathbb{R}^d, \mathbb{R}^d)\) or in \(L_{\mathrm{loc}}^{2+\varepsilon}(\mathbb{R}^d, \mathbb{R}^d)\) for some \(\varepsilon >0\), while \(\rho\) is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.Analysis of an elastic-rigid obstacle problem described by a variational-hemivariational inequalityhttps://zbmath.org/1540.490132024-09-13T18:40:28.020319Z"Wang, Xilu"https://zbmath.org/authors/?q=ai:wang.xilu"Ran, Qinghua"https://zbmath.org/authors/?q=ai:ran.qinghua"Xiao, Qichang"https://zbmath.org/authors/?q=ai:xiao.qichangThe authors consider an obstacle problem for the elastic membrane lying above an elastic-rigid obstacle. The elastic-rigid obstacle allows limited penetration and offers a nonmonotone reactive force. They introduce the mathematical model and prove that its weak form, which is a variational-hemivariational inequality, has a unique solution. Then, a discrete scheme is adopted to solve the problem. The optimal-order error estimate under appropriate regularity assumptions is derived. Finally, numerical examples are reported, from which the theoretical predicted optimal-order error estimate can be clearly obtained.
Reviewer: Zijia Peng (Nanning)Geometric criteria for the existence of capillary surfaces in tubeshttps://zbmath.org/1540.490292024-09-13T18:40:28.020319Z"Saracco, Giorgio"https://zbmath.org/authors/?q=ai:saracco.giorgioThe author proves some geometric criteria that yield existence of capillary surfaces in tubes \(\Omega \times \mathbb{R}\) in a gravity free environment, where \(\Omega \subset \mathbb{R}^{2}\) is bounded, open, and simply connected. Up to the multiplicative factor of the surface tension, the energy of the system is \(\int_{\Omega }\sqrt{1+\left\vert \nabla u\right\vert ^{2}}dx-\cos(\gamma )\int_{\partial \Omega }ud\mathcal{H} ^{1}(x)+\int_{\Omega }\lambda udx\), where the first term is the surface energy, the second one the adhesion energy, \(\gamma \) being the contact angle measured inside the lower fluid between the phases and the cylinder, and the last one represents the volume constraint, \(\lambda \) being a Lagrange multiplier. Writing the Euler-Lagrange equation of the energy functional, smooth critical points need to satisfy : \(\operatorname{div}(Tu)=\lambda \), in \( \Omega \), \(Tu\cdot \nu _{\Omega }=\cos(\gamma )\), on \(\partial \Omega \), where \(Tu=\nabla u/\sqrt{1+\left\vert \nabla u\right\vert ^{2}}\), and \(\nu _{\Omega }\) the outward normal to \(\Omega \). Applying the Gauss-Green Theorem to this Euler-Lagrange equation and assuming \(\Omega \) to be Lipschitz, the author derives: \(-\int_{\partial \Omega }\frac{\nabla u\cdot \nu _{\Omega }}{\sqrt{1+\left\vert \nabla u\right\vert ^{2}}}d\mathcal{H} ^{1}(x)=\int_{\Omega }\operatorname{div}(Tu)dx=\lambda \int_{\Omega }1dx=\lambda \left\vert \Omega \right\vert \), which leads to the necessary condition to existence \( \lambda \leq \frac{P(\Omega )}{\left\vert \Omega \right\vert }\), where \( P(\Omega )\) denotes the distributional perimeter, which for Lipschitz sets \(E \) coincides with \(\mathcal{H}^{1}(\partial E)\). It is possible to refine this condition as \(\lambda =\cos(\gamma )\frac{P(\Omega )}{\left\vert \Omega \right\vert }\) and to perform the same reasoning on any proper Lipschitz subset \(E\) of \(\Omega \), leading to the necessary condition to existence \( \lambda <\frac{P(E;\Omega )+\cos(\gamma )P(E;\partial \Omega )}{\left\vert E\right\vert }\), where \(P(E;A)\) denotes the perimeter of \(E\) relative to the set \(A\). Considering the more general problem \(\operatorname{div}(Tu)=H\), in \(\Omega \), where \(H\) is a fixed positive constant, the author derives the necessary condition to existence \(H<\frac{P(E)}{\left\vert E\right\vert }\), for all proper subsets \(E\subset \Omega \) of locally finite perimeter. Given a bounded, open, and simply connected subset \(\Omega \) of \(\mathbb{R}^{2}\) satisfying \(P(\Omega )=\mathcal{H}^{1}(\partial \Omega )\) and the Poincaré-type inequality for all subsets \(E\subset \Omega \): \(\min\{P(E;\partial \Omega );P(\Omega \setminus E;\partial \Omega )\}\leq kP(E;\Omega)\), for some positive constant \(k\) depending only on \(\Omega \), and \(H\in \mathbb{R}\) fixed, the three assertions are equivalent: The inequality \(H< \frac{P(E)}{\left\vert E\right\vert }\) holds and \(H=\frac{P(\Omega )}{ \left\vert \Omega \right\vert }\); There exists a unique (up to translations) solution of \(\operatorname{div}(Tu)=H\), in \(\Omega \); There exists a solution \(u\) such that \(Tu\cdot \nu _{\Omega }=1\) a.e. on \(\partial \Omega \), i.e. that solves \( \operatorname{div}(Tu)=\lambda \), in \(\Omega \), \(Tu\cdot \nu _{\Omega }=\cos(\gamma )\), on \( \partial \Omega \), with \(\gamma =0\). The first criterion deals with convex sets.\ Let \(\Omega \subset \mathbb{R}^{2}\) be a bounded, open, and convex set. Then the problem \(\operatorname{div}(Tu)=\lambda \), in \(\Omega \), \(Tu\cdot \nu _{\Omega }=\cos(\gamma )\), on \(\partial \Omega \) has a solution for \(\gamma =0 \) if and only if \(\overline{\kappa }=\mathrm{ess}\ \sup \ \kappa _{\partial \Omega }\leq \frac{P(\Omega )}{\left\vert \Omega \right\vert }\), where \(\kappa _{\partial \Omega }\) represents the curvature of \(\partial \Omega \). This is derived from the preceding result, the author observing that convexity implies Lipschitz continuity. Moving to the case of non convex piecewise Lipschitz sets, the author introduces the strict interior rolling ball condition of radius \(r\) and of absence of necks of radius \(r\). If \(\Omega \subset \mathbb{ R}^{2}\) is a bounded, piecewise Lipschitz, and simply connected set which enjoys the strict interior rolling ball condition for \(r=\frac{\left\vert \Omega \right\vert }{P(\Omega )}\), then the problem \(\operatorname{div}(Tu)=\lambda \), in \( \Omega \), \(Tu\cdot \nu _{\Omega }=\cos(\gamma )\), on \(\partial \Omega \), has a solution for \(\gamma =0\). If \(\Omega \) has no necks of radius \(r=\frac{ \left\vert \Omega \right\vert }{P(\Omega )}\), then the problem \( \operatorname{div}(Tu)=\lambda \), in \(\Omega \), \(Tu\cdot \nu _{\Omega }=\cos(\gamma )\), on \( \partial \Omega \), has a solution for \(\gamma =0\) if and only if \(\Omega \) enjoys the strict interior rolling ball condition for \(r=\frac{\left\vert \Omega \right\vert }{P(\Omega )}\). Finally, the author proves a criterion in the case of a Jordan set \(\Omega \).
Reviewer: Alain Brillard (Riedisheim)Plateau's problem via the Allen-Cahn functionalhttps://zbmath.org/1540.490462024-09-13T18:40:28.020319Z"Guaraco, Marco A. M."https://zbmath.org/authors/?q=ai:guaraco.marco-a-m"Lynch, Stephen"https://zbmath.org/authors/?q=ai:lynch.stephenSummary: Let \(\Gamma\) be a compact codimension-two submanifold of \(\mathbb{R}^n\), and let \(L\) be a nontrivial real line bundle over \(X = \mathbb{R}^n \setminus \Gamma\). We study the Allen-Cahn functional,
\[
E_\varepsilon(u) = \int_X \varepsilon\frac{|\nabla u|^2}{2} + \frac{(1-|u|^2)^2}{4\varepsilon}dx,
\]
on the space of sections \(u\) of \(L\). Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to \(\Gamma\). We first show that, for a family of critical sections with uniformly bounded energy, in the limit as \(\varepsilon\rightarrow 0\), the associated family of energy measures converges to an integer rectifiable \((n-1)\)-varifold \(V\). Moreover, \(V\) is stationary with respect to any variation which leaves \(\Gamma\) fixed. Away from \(\Gamma\), this follows from work of Hutchinson-Tonegawa; our result extends their interior theory up to the boundary \(\Gamma\). Under additional hypotheses, we can say more about \(V\). When \(V\) arises as a limit of critical sections with uniformly bounded Morse index, \(\Sigma := \mathrm{supp}\|V\|\) is a minimal hypersurface, smooth away from \(\Gamma\) and a singular set of Hausdorff dimension at most \(n - 8\). If the sections are globally energy minimizing and \(n = 3\), then \(\Sigma\) is a smooth surface with boundary, \(\partial\Sigma = \Gamma\) (at least if \(L\) is chosen correctly), and \(\Sigma\) has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich and Struwe) that the smooth version of Plateau's problem admits a solution for every boundary curve in \(\mathbb{R}^3\). This also works if \(4 \leq n \leq 7\) and \(\Gamma\) is assumed to lie in a strictly convex hypersurface.The class of second fundamental forms arising from minimal immersions in a space formhttps://zbmath.org/1540.530792024-09-13T18:40:28.020319Z"Lucia, Marcello"https://zbmath.org/authors/?q=ai:lucia.marcelloSummary: The second fundamental form arising from an oriented minimal immersion of a closed surface in a space form satisfies several constraints. One of them is provided by the Gauss-Codazzi equation that can be rephrased as a semilinear problem on the surface. We discuss some results for these type of nonlinear problems and analyze the behaviors of the solutions when the hyperbolic norm of the second fundamental form is small.Flow by Gauss curvature to the \(L_p\) dual Minkowski problemhttps://zbmath.org/1540.531132024-09-13T18:40:28.020319Z"Guang, Qiang"https://zbmath.org/authors/?q=ai:guang.qiang"Li, Qi-Rui"https://zbmath.org/authors/?q=ai:li.qirui"Wang, Xu-Jia"https://zbmath.org/authors/?q=ai:wang.xu-jiaThe theory nowadays known as Brunn-Minkowski theory concerns the study of geometric functionals defined from convex bodies. A Minkowski problem is a characterization problem for a geometric measure generated by convex bodies. Its solution amounts to solving a degenerate fully nonlinear partial differential equation, see for instance [\textit{Y. Huang} et al., Acta Math. 216, No. 2, 325--388 (2016; Zbl 1372.52007)].
Let \(\mathcal M_0\) be a smooth closed uniformly convex hypersurface in \(\mathbb R^{n+1}\) enclosing the origin. Consider the Gauss curvature flow
\[
\begin{cases} \partial_t X(x,t)=-f(v)r^{\alpha}K(x,t)v,\\
X(x,0)=X_0(x), \end{cases}
\]
where
\begin{itemize}
\item[1.] \(K(\cdot,t)\) is the Gauss curvature of the hypersurface \(\mathcal M_t\) parameterized by \(X(\cdot,t):\mathbb S^{n}\rightarrow \mathbb R^{n+1} \),
\item[2.] \(v(\cdot,t)\) is the unit outer normal vector at \(X(\cdot,t)\),
\item[3.] \(f\) is a given smooth positive function on \(\mathbb{S}^n\),
\item[4.] \(r=|X(x,t)|\) is the distance of \(X(x,t)\) to the origin.
\end{itemize}
The flow above was introduced to study the existence of solutions for the dual Minkowski problem proposed in [Y. Huang et al., loc. cit.]. This turns out to be equivalent to the following Monge-Ampère problem on \(\mathbb{S}^n\):
\[
\mathrm{det}\left(\nabla^2u+uI\right)=\frac{f(x)}{u}\left(|\nabla u(x)|^2{+}u^2\right)^{\alpha/2},
\]
where \(u\) denotes the support function of a hypersurface solution \(\mathcal{M}\).
The \(L_{p}\)-Minkowski problem, herein LpM, introduced in [\textit{E. Lutwak}, J. Differ. Geom. 38, No. 1, 131--150 (1993; Zbl 0788.52007)], concerns the existence of closed convex hypersurfaces with prescribed \(p\)-area measure. By adapting the former Monge-Ampère equation, varying parameters include the dual problem to LpM. That is,
\[
\mathrm{det}\left(\nabla^2u+uI\right)=\frac{f(x)u^{p-1}}{g\left(\frac{\nabla u(x)+ux}{\sqrt{|\nabla u(x)|^2+u(x)^2}} \right)}\left(|\nabla u(x)|^2{+}u^2\right)^{(n+1-q)/2}.
\]
This contains the dual LpM problem for \(g\equiv 1\).
The main result in the present paper provides sufficient existence conditions for this adapted Monge-Ampère problem assuming \(\alpha \in (0,1)\) and some regularity assumptions on \(f, g\) under constraints on \(p,q\).
The proof of the main result is achieved using the Gauss flow given above.
Reviewer: Leonardo Francisco Cavenaghi (Campinas)A mountain pass algorithm for quasilinear boundary value problem with \(p\)-Laplacianhttps://zbmath.org/1540.580092024-09-13T18:40:28.020319Z"Bailová, Michaela"https://zbmath.org/authors/?q=ai:bailova.michaela"Bouchala, Jiří"https://zbmath.org/authors/?q=ai:bouchala.jiriSummary: In this paper, we deal with a specific type of quasilinear boundary value problem with Dirichlet boundary conditions and with \(p\)-Laplacian. We show two ways of proving the existence of nontrivial weak solutions. The first one uses the mountain pass theorem, the other one is based on our new minimax theorem. This method is novel even for \(p = 2\). In the paper, we also present a numerical algorithm based on the introduced approach. The suggested algorithm is illustrated on numerical examples and compared with a current approach to demonstrate its efficiency.On large deviation principles and the Monge-Ampère equation (following Berman, Hultgren)https://zbmath.org/1540.600462024-09-13T18:40:28.020319Z"Rubinstein, Yanir A."https://zbmath.org/authors/?q=ai:rubinstein.yanir-aSummary: This is mostly an exposition, aimed to be accessible to geometers, analysts, and probabilists, of a fundamental recent theorem of R. Berman with recent developments by J. Hultgren, that asserts that the second boundary value problem for the real Monge-Ampère equation admits a probabilistic interpretation, in terms of many particle limit of permanental point processes satisfying a large deviation principle with a rate function given explicitly using optimal transport. An alternative proof of a step in the Berman-Hultgren Theorem is presented allowing to to deal with all ``tempratures'' simultaneously instead of first reducing to the zero-temperature case.
For the entire collection see [Zbl 1515.14010].Shape of eigenvectors for the decaying potential modelhttps://zbmath.org/1540.601542024-09-13T18:40:28.020319Z"Nakano, Fumihiko"https://zbmath.org/authors/?q=ai:nakano.fumihikoSummary: We consider the 1d Schrödinger operator with decaying random potential and study the joint scaling limit of the eigenvalues and the measures associated with the corresponding eigenfunctions, which is based on the formulation by \textit{B. Rifkind} and \textit{B. Virág} [Geom. Funct. Anal. 28, No. 5, 1394--1419 (2018; Zbl 1459.60013)]. As a result, we have completely different behavior depending on the decaying rate \(\alpha > 0\) of the potential: The limiting measure is equal to (1) Lebesgue measure for the supercritical case \((\alpha > 1/2)\), (2) a measure of which the density has power-law decay with Brownian fluctuation for critical case \((\alpha =1/2)\), and (3) the delta measure with its atom being uniformly distributed for the subcritical case \((\alpha <1/2)\). This result is consistent with the previous study on spectral and statistical properties.Efficient simulation of the Schrödinger equation with a piecewise constant positive potentialhttps://zbmath.org/1540.601862024-09-13T18:40:28.020319Z"Yang, Xuxin"https://zbmath.org/authors/?q=ai:yang.xuxin"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.antti"Sottinen, Tommi"https://zbmath.org/authors/?q=ai:sottinen.tommiSummary: We introduce a new method for the Monte Carlo simulation of a weak solution of the Schrödinger-type equation where the potential is piecewise constant and positive. The method, called the killing walk-on-spheres algorithm, combines the classical walk-on-spheres algorithm with killing that can be determined by using panharmonic measures. This paper continues our earlier work in which simulation of the solutions of the Yukawa and the Helmholtz partial differential equations were developed.Scaling limit for a second-order particle system with local annihilationhttps://zbmath.org/1540.602182024-09-13T18:40:28.020319Z"Huang, Ruojun"https://zbmath.org/authors/?q=ai:huang.ruojunSummary: For a second-order particle system in \(\mathbb{R}^d\) subject to locally-in-space pairwise annihilation, we prove a scaling limit for its empirical measure on position and velocity towards a degenerate elliptic partial differential equation. Crucial ingredients are Green's function estimates for the associated hypoelliptic operator and an Itô-Tanaka trick.Corrigendum to: ``Quasi-Monte Carlo finite element analysis for wave propagation in heterogeneous random media''https://zbmath.org/1540.650172024-09-13T18:40:28.020319Z"Ganesh, M."https://zbmath.org/authors/?q=ai:ganesh.mahadevan"Kuo, Frances Y."https://zbmath.org/authors/?q=ai:kuo.frances-y"Sloan, Ian H."https://zbmath.org/authors/?q=ai:sloan.ian-hFrom the text: Our paper [ibid. 9, 106--134 (2021; Zbl 1459.65010)] analyzes wave propagation in a medium for which the refractive index is a random field. The paper has an overlooked dependence on the wavenumber \(k\) in the error bound of the finite-element approximation of the governing partial differential equation (PDE). This additional wavenumber dependence needs to be inserted into the finite-element contribution to the total error in Theorems 7.1 and 7.2 and in the corresponding statement of total error in the introduction of the paper.Finite sections: a functional analytic perspective on approximation methodshttps://zbmath.org/1540.651622024-09-13T18:40:28.020319Z"Lindner, Marko"https://zbmath.org/authors/?q=ai:lindner.marko"Seifert, Christian"https://zbmath.org/authors/?q=ai:seifert.christianSummary: We review approximation methods and their stability and applicability. We then focus on the finite section method and Galerkin methods and show that on separable Hilbert spaces either one can be interpreted as the other. In the end we demonstrate that well-known methods such as the finite element method and polynomial chaos expansion are particular examples of the finite section method; their applicability can therefore be studied via the latter.
{\copyright} 2018 WILEY-VCH Verlag GmbH \& Co. KGaA, WeinheimIterative PDE-constrained optimization for seismic full-waveform inversionhttps://zbmath.org/1540.651732024-09-13T18:40:28.020319Z"Malovichko, M. S."https://zbmath.org/authors/?q=ai:malovichko.m-s"Orazbayev, A."https://zbmath.org/authors/?q=ai:orazbayev.a"Khokhlov, N. I."https://zbmath.org/authors/?q=ai:khokhlov.nikolai-i|khokhlov.nikolay-i"Petrov, I. B."https://zbmath.org/authors/?q=ai:petrov.igor-borisovichSummary: This paper presents a novel numerical method for the Newton seismic full-waveform inversion (FWI). The method is based on the full-space approach, where the state, adjoint state, and control variables are optimized simultaneously. Each Newton step is formulated as a PDE-constrained optimization problem, which is cast in the form of the Karush-Kuhn-Tucker (KKT) system of linear algebraic equitations. The KKT system is solved inexactly with a preconditioned Krylov solver. We introduced two preconditioners: the one based on the block-triangular factorization and its variant with an inexact block solver. The method was benchmarked against the standard truncated Newton FWI scheme on a part of the Marmousi velocity model. The algorithm demonstrated a considerable runtime reduction compared to the standard FWI. Moreover, the presented approach has a great potential for further acceleration. The central result of this paper is that it establishes the feasibility of Newton-type optimization of the KKT system in application to the seismic FWI.Lower bounds, elliptic reconstruction and a posteriori error control of parabolic problemshttps://zbmath.org/1540.653742024-09-13T18:40:28.020319Z"Georgoulis, Emmanuil H."https://zbmath.org/authors/?q=ai:georgoulis.emmanuil-h"Makridakis, Charalambos G."https://zbmath.org/authors/?q=ai:makridakis.charalambos-gSummary: A popular approach for proving \textit{a posteriori} error bounds in various norms for evolution problems with partial differential equations uses \textit{reconstruction operators} to recover conforming objects from the approximate solutions. So far, lower bounds in reconstruction-based \textit{a posteriori} error estimators have been proven only for time-discrete schemes for parabolic problems; the proof of lower bounds for \textit{fully discrete} schemes in reconstruction-based \textit{a posteriori} error estimators has eluded. In this work, we provide a complete framework addressing this issue for energy-type norms. We consider Backward Euler discretizations and time-discontinuous Galerkin schemes, combined with dynamically changing conforming finite element methods in space, approximating linear parabolic problems. The results presented include sharp upper and lower \textit{a posteriori} error bounds. Localized versions of the lower bounds are also considered.An investigation of global radial basis function collocation methods applied to Helmholtz problemshttps://zbmath.org/1540.654212024-09-13T18:40:28.020319Z"Larsson, Elisabeth"https://zbmath.org/authors/?q=ai:larsson.elisabeth"Sundin, Ulrika"https://zbmath.org/authors/?q=ai:sundin.ulrikaSummary: Global radial basis function (RBF) collocation methods with infinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution functions. At the same time, the linear systems that arise are dense and severely ill-conditioned for large numbers of unknowns and small values of the shape parameter that determines how flat the basis functions are. We use Helmholtz equation as an application problem for the theoretical analysis and numerical experiments. We analyze and characterize the convergence properties as a function of the number of unknowns and for different shape parameter ranges. We provide theoretical results for the flat limit of the PDE solutions and investigate when the non-symmetric collocation matrices become singular. We also provide practical strategies for choosing the method parameters and evaluate the results on Helmholtz problems in a curved waveguide geometry.Distributed source scheme to solve the classical form of Poisson equation using 3-d finite-difference method for improved accuracy and unrestricted source positionhttps://zbmath.org/1540.654542024-09-13T18:40:28.020319Z"Goona, Nithin Kumar"https://zbmath.org/authors/?q=ai:goona.nithin-kumar"Parne, Saidi Reddy"https://zbmath.org/authors/?q=ai:parne.saidi-reddy"Sashidhar, S."https://zbmath.org/authors/?q=ai:sashidhar.sSummary: The Finite-Difference Method (FDM) despite being old and simple is not being used as rigorously as its counterpart Finite Element Method (FEM) for solving partial differential equations. This study aims to examine and improve the accuracy of FDM by eliminating significant sources of error. Since an expression for exact potential from the most accurate Method of Moment (MoM) is available in 3-D electrostatics, the classical form of Poisson equation is chosen for this study such that the error due to boundary conditions can be eliminated. The error due to the source term in the Poisson equation is studied with a single source and different grid densities by applying FDM in 3-D. Since the error is only present in the immediate surroundings of the source, a Distributed Source Scheme (DSS) has been proposed to reduce error due to the source term. A modified DSS i.e., Truncated Distributed Source Scheme (TDSS) is applied for the practical implementation of DSS. The maximum error in FDM when the source term is present has been reduced from 8.151\% to 0.00091\% with DSS. With the application of TDSS, it is shown that the maximum error can be maintained well below 0.1\% for truncation values \(n > 15\). The error due to source at off-center and off-grid positions was computed using TDSS and the maximum error is observed to be less than 0.05\% and 0.01\%, respectively. With off-grid error being low due to TDSS, it is shown that sources in TDSS can now take any position irrespective of grid nodes, which is forbidden in FDM with an average maximum error of 0.026\%. It is also shown that DSS can also be used to find the charge distribution for a given potential distribution. While still maintaining the simplicity, improved accuracy and unrestricted source positions are achieved in FDM with exact boundary conditions using DSS.The immersed interface method for Helmholtz equations with degenerate diffusionhttps://zbmath.org/1540.654582024-09-13T18:40:28.020319Z"Medina Dorantes, Francisco"https://zbmath.org/authors/?q=ai:medina-dorantes.francisco"Itzá Balam, Reymundo"https://zbmath.org/authors/?q=ai:itzabalam.reymundo|itza-balam.reymundo"Uh Zapata, Miguel"https://zbmath.org/authors/?q=ai:zapata.miguel-uh|uhzapata.miguelSummary: In this paper, we consider a second-order immersed interface method for Helmholtz equations of the form \(\nabla ( \beta \nabla u ) - \sigma u = f\) with a degenerate diffusion term \(\beta\). We assume that the diffusion term is discontinuous across an interface and \(\beta\) is zero to one side of it. The method is applied to one-dimensional domains with multiple interfaces, and two-dimensional domains with circular and straight interfaces. The numerical solution is obtained by applying away from the interface the standard centered finite differences scheme and a new scheme across of the interface. Numerical results on one- and two-dimensional domains are used to compare and demonstrate the proposed numerical method's capabilities. In all numerical experiments, the solutions of the interface problem is second order of accuracy.A compact sixth-order implicit immersed interface method to solve 2D Poisson equations with discontinuitieshttps://zbmath.org/1540.654612024-09-13T18:40:28.020319Z"Uh Zapata, M."https://zbmath.org/authors/?q=ai:uh-zapata.m-a|uhzapata.miguel"Itza Balam, R."https://zbmath.org/authors/?q=ai:itza-balam.reymundo"Montalvo-Urquizo, J."https://zbmath.org/authors/?q=ai:montalvo-urquizo.jonathanSummary: This paper proposes a compact sixth-order accurate numerical method to solve Poisson equations with discontinuities across an interface. This scheme is based on two techniques for the second-order derivative approximation: a high-order implicit finite difference (HIFD) formula to increase the precision and an immersed interface method (IIM) to deal with the discontinuities. The HIFD formulation arises from Taylor series expansion, and the new formulas are simple modifications to the standard finite difference schemes. On the other hand, the IIM allows one to solve the differential equation using a fixed Cartesian grid by adding some correction terms only at grid points near the immersed interface. The two-dimensional equation is then solved by a nine-point compact sixth-order scheme named HIFD-IIM. Fourth- and second-order methods result in particular cases of the proposed method. Furthermore, the sixth-order method requires similar computational resources to a fourth-order formulation because the resulting matrices in both discretizations are the same. However, higher-order methods require the knowledge of more jump conditions at the interface. From the theoretical derivation of the proposed method, we expect fully six-order accuracy in the maximum norm. This order has been confirmed from our numerical experiments using nontrivial analytical solutions.Benchmarking the geometrical robustness of a virtual element Poisson solverhttps://zbmath.org/1540.654722024-09-13T18:40:28.020319Z"Attene, Marco"https://zbmath.org/authors/?q=ai:attene.marco"Biasotti, Silvia"https://zbmath.org/authors/?q=ai:biasotti.silvia"Bertoluzza, Silvia"https://zbmath.org/authors/?q=ai:bertoluzza.silvia"Cabiddu, Daniela"https://zbmath.org/authors/?q=ai:cabiddu.daniela"Livesu, Marco"https://zbmath.org/authors/?q=ai:livesu.marco"Patanè, Giuseppe"https://zbmath.org/authors/?q=ai:patane.giuseppe"Pennacchio, Micol"https://zbmath.org/authors/?q=ai:pennacchio.micol"Prada, Daniele"https://zbmath.org/authors/?q=ai:prada.daniele"Spagnuolo, Michela"https://zbmath.org/authors/?q=ai:spagnuolo.michelaSummary: Polytopal Element Methods (PEM) allow us solving differential equations on general polygonal and polyhedral grids, potentially offering great flexibility to mesh generation algorithms. Differently from classical finite element methods, where the relation between the geometric properties of the mesh and the performances of the solver are well known, the characterization of a good polytopal element is still subject to ongoing research. Current shape regularity criteria are quite restrictive, and greatly limit the set of valid meshes. Nevertheless, numerical experiments revealed that PEM solvers can perform well on meshes that are far outside the strict boundaries imposed by the current theory, suggesting that the real capabilities of these methods are much higher. In this work, we propose a benchmark to study the correlation between general 2D polygonal meshes and PEM solvers which we test on a virtual element solver for the Poisson equation. The benchmark aims to explore the space of 2D polygonal meshes and polygonal quality metrics, in order to understand if and how shape regularity, defined according to different criteria, affects the performance of the method. The proposed tool is quite general, and can be potentially used to study any PEM solver. Besides discussing the basics of the benchmark, we demonstrate its application on a representative member of the PEM family, namely the Virtual Element Method, also discussing our findings.Novel design and analysis of generalized finite element methods based on locally optimal spectral approximationshttps://zbmath.org/1540.654902024-09-13T18:40:28.020319Z"Ma, Chupeng"https://zbmath.org/authors/?q=ai:ma.chupeng"Scheichl, Robert"https://zbmath.org/authors/?q=ai:scheichl.robert"Dodwell, Tim"https://zbmath.org/authors/?q=ai:dodwell.tim-jSummary: In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a partition of unity are presented. These new spaces have advantages over those proposed in [\textit{I. Babuska} and \textit{R. Lipton}, Multiscale Model. Simul. 9, No. 1, 373--406 (2011; Zbl 1229.65195)]. First, in addition to a nearly exponential decay rate of the local approximation errors with respect to the dimensions of the local spaces, the rate of convergence with respect to the size of the oversampling region is also established. Second, the theoretical results hold for problems with mixed boundary conditions defined on general Lipschitz domains. Finally, an efficient and easy-to-implement technique for generating the discrete \(A\)-harmonic spaces is proposed which relies on solving an eigenvalue problem associated with the Dirichlet-to-Neumann operator, leading to a substantial reduction in computational cost. Numerical experiments are presented to support the theoretical analysis and to confirm the effectiveness of the new method.Stochastic finite differences for elliptic diffusion equations in stratified domainshttps://zbmath.org/1540.654912024-09-13T18:40:28.020319Z"Maire, Sylvain"https://zbmath.org/authors/?q=ai:maire.sylvain"Nguyen, Giang"https://zbmath.org/authors/?q=ai:nguyen.giang-v|nguyen.giang-linh-duc|nguyen.giang-thu|nguyen.giang-dSummary: We describe Monte Carlo algorithms to solve elliptic partial differential equations with piecewise constant diffusion coefficients and general boundary conditions including Robin and transmission conditions as well as a damping term. The treatment of the boundary conditions is done via stochastic finite differences techniques which possess a higher order than the usual methods. The simulation of Brownian paths inside the domain relies on variations around the walk on spheres method with or without killing. We check numerically the efficiency of our algorithms on various examples of diffusion equations illustrating each of the new techniques introduced here.An analysis of nonconforming virtual element methods on polytopal meshes with small faceshttps://zbmath.org/1540.654932024-09-13T18:40:28.020319Z"Park, Hyeokjoo"https://zbmath.org/authors/?q=ai:park.hyeokjoo"Kwak, Do Y."https://zbmath.org/authors/?q=ai:kwak.do-youngSummary: In this paper, we analyze nonconforming virtual element methods on polytopal meshes with small faces for the second-order elliptic problem. We propose new stability forms for 2D and 3D nonconforming virtual element methods. For the 2D case, the stability form is defined by the sum of an inner product of approximate tangential derivatives and a weighed \(L^2\)-inner product of certain projections on the mesh element boundaries. For the 3D case, the stability form is defined by a weighted \(L^2\)-inner product on the mesh element boundaries. We prove the optimal convergence of the nonconforming virtual element methods equipped with such stability forms. Finally, several numerical experiments are presented to verify our analysis and compare the performance of the proposed stability forms with the standard stability form
[\textit{B. Ayuso de Dios} et al., ESAIM, Math. Model. Numer. Anal. 50, No. 3, 879--904 (2016; Zbl 1343.65140)].Uniform convergence of a weak Galerkin method for singularly perturbed convection-diffusion problemshttps://zbmath.org/1540.654992024-09-13T18:40:28.020319Z"Zhang, Jin"https://zbmath.org/authors/?q=ai:zhang.jin.1"Liu, Xiaowei"https://zbmath.org/authors/?q=ai:liu.xiaoweiSummary: In this article, we analyze convergence of a weak Galerkin method on Bakhvalov-type mesh. This method uses piecewise polynomials of degree \(k \geq 1\) on the interior and piecewise constant on the boundary of each element. To obtain uniform convergence, we carefully define the penalty parameter and a new interpolant which is based on the characteristic of the Bakhvalov-type mesh. Then the method is proved to be convergent with optimal order, which is confirmed by numerical experiments.Geometrically transformed spectral methods to solve partial differential equations in circular geometries, application for multi-phase flowhttps://zbmath.org/1540.655022024-09-13T18:40:28.020319Z"Assar, Moein"https://zbmath.org/authors/?q=ai:assar.moein"Grimes, Brian Arthur"https://zbmath.org/authors/?q=ai:grimes.brian-arthurSummary: Circular geometries are ubiquitously encountered in science and technology, and the polar coordinate provides the natural way to analyze them; However, its application is limited to symmetric cases, and it cannot be applied to segments that are formed in multiphase flow problems in pipes. To address that, spectral discretization of circular geometries via orthogonal collocation technique is developed using geometrical mapping. Two analytical mappings between the circle and square geometries, namely, elliptical and horizontally squelched mappings, are employed. Accordingly, numerical algorithms are developed for solving PDEs in circular geometries with different boundary conditions for both steady state and transient problems. Various implementation issues are thoroughly discussed, including vectorization and strategies to avoid solving the differential-algebraic system of equations. Moreover, several case studies for symmetric and asymmetric Poisson equations with different boundary conditions are performed to evaluate several aspects of these techniques, such as error properties, condition number, and computational time. For both steady state and transient solvers, it was revealed that the computation time scales quadratically with respect to the grid size for both mapping and polar discretization techniques. However, due to the presence of the second mixed derivative, mapping techniques are more computationally costly. Finally, the squelched mapping was successfully employed to discretize the two, and three-phase gravity flows in sloped pipes.On solving elliptic boundary value problems using a meshless method with radial polynomialshttps://zbmath.org/1540.655072024-09-13T18:40:28.020319Z"Ku, Cheng-Yu"https://zbmath.org/authors/?q=ai:ku.cheng-yu"Xiao, Jing-En"https://zbmath.org/authors/?q=ai:xiao.jing-en"Liu, Chih-Yu"https://zbmath.org/authors/?q=ai:liu.chih-yu"Lin, Der-Guey"https://zbmath.org/authors/?q=ai:lin.der-gueySummary: This paper presents the meshless method using radial polynomials with the combination of the multiple source collocation scheme for solving elliptic boundary value problems. In the proposed method, the basis function is based on the radial polynomials, which is different from the conventional radial basis functions that approximate the solution using the specific function such as the multiquadric function with the shape parameter for infinitely differentiable. The radial polynomial basis function is a non-singular series function in nature which is infinitely smooth and differentiable in nature without using the shape parameter. With the combination of the multiple source collocation scheme, the center point is regarded as the source point for the interpolation of the radial polynomials. Numerical solutions in multiple dimensions are approximated by applying the radial polynomials with given terms of the radial polynomials. The comparison of the proposed method with the radial basis function collocation method (RBFCM) using the multiquadric and polyharmonic spline functions is conducted. Results demonstrate that the accuracy obtained from the proposed method is better than that of the conventional RBFCM with the same number of collocation points. In addition, highly accurate solutions with the increase of radial polynomial terms may be obtained.The MAPS based on trigonometric basis functions for solving elliptic partial differential equations with variable coefficients and Cauchy-Navier equationshttps://zbmath.org/1540.655102024-09-13T18:40:28.020319Z"Wang, Dan"https://zbmath.org/authors/?q=ai:wang.dan.3"Chen, C. S."https://zbmath.org/authors/?q=ai:chen.ching-shyang"Fan, C. M."https://zbmath.org/authors/?q=ai:fan.chia-ming|fan.cheng-ming|fan.congmin|fan.chengmei|fan.chen-ming"Li, Ming"https://zbmath.org/authors/?q=ai:li.ming.10Summary: In this paper, we extended the method of approximate particular solutions (MAPS) using trigonometric basis functions to solve two-dimensional elliptic partial differential equations (PDEs) with variable-coefficients and the Cauchy-Navier equations. The new approach is based on the closed-form particular solutions for second-order differential operators with constant coefficients. For the Cauchy-Navier equations, a reformulation of the equations is required so that the particular solutions for the new differential operators are available. Five numerical examples are provided to demonstrate the effectiveness of the proposed method.The localized method of fundamental solutions for 2D and 3D inhomogeneous problemshttps://zbmath.org/1540.655142024-09-13T18:40:28.020319Z"Zhang, Junli"https://zbmath.org/authors/?q=ai:zhang.junli"Yang, Chenchen"https://zbmath.org/authors/?q=ai:yang.chenchen"Zheng, Hui"https://zbmath.org/authors/?q=ai:zheng.hui"Fan, Chia-Ming"https://zbmath.org/authors/?q=ai:fan.chia-ming"Fu, Ming-Fu"https://zbmath.org/authors/?q=ai:fu.mingfuSummary: In this paper, the newly-developed localized method of fundamental solutions (LMFS) is extended to analyze multi-dimensional boundary value problems governed by inhomogeneous partial differential equations (PDEs). The LMFS can acquire highly accurate numerical results for the homogeneous PDEs with an incredible improvement of the computational speed. However, the LMFS cannot be directly used for inhomogeneous PDEs. Traditional two-steps scheme has difficulties in finding the particular solutions, and will lead to a loss of the accuracy and efficiency. In this paper, the recursive composite multiple reciprocity method (RC-MRM) is adopted to re-formulate the inhomogeneous PDEs to higher-order homogeneous PDEs with additional boundary conditions, which can be solved by the LMFS efficiently and accurately. The proposed combination of the RC-MRM and the LMFS can analyze inhomogeneous governing equation directly and avoid troublesome caused by the two-steps schemes. The details of the numerical discretization of the RC-MRM and the LMFS are elaborated. Some numerical examples are provided to demonstrate the accuracy and efficiency of the proposed scheme. Furthermore, some key factors of the LMFS are systematically investigated to show the merits of the proposed meshless scheme.Monte Carlo solution of the Neumann problem for the nonlinear Helmholtz equationhttps://zbmath.org/1540.655252024-09-13T18:40:28.020319Z"Rasulov, Abdujabar"https://zbmath.org/authors/?q=ai:rasulov.abdujabar"Raimova, Gulnora"https://zbmath.org/authors/?q=ai:raimova.gulnoraSummary: In this paper we will consider the Neumann boundary-value problem for the Helmholtz equation with a polynomial nonlinearity on the right-hand side. We will assume that a solution to our problem exists, and this permits us to construct an unbiased Monte Carlo estimator using the trajectories of certain branching processes. To do so we utilize Green's formula and an elliptic mean-value theorem. This allows us to derive a special integral equation, which equates the value of the function \(u(x)\) at the point \(x\) with its integral over the domain \(D\) and on boundary of the domain \(\partial D = G\). The solution of the problem is then given in the form of a mathematical expectation over some particular random variables. According to this probabilistic representation, a branching stochastic process is constructed and an unbiased estimator of the solution of the nonlinear problem is formed by taking the expectation over this branching process. The unbiased estimator which we derive has a finite variance. In addition, the proposed branching process has a finite average number of branches, and hence is easily simulated. Finally, we provide numerical results based on numerical experiments carried out with these algorithms to validate our approach.On some results about the variational theory of complex rays used close to the high frequency regimehttps://zbmath.org/1540.740602024-09-13T18:40:28.020319Z"Riou, Hervé"https://zbmath.org/authors/?q=ai:riou.herveSummary: This work presents some results about the variational theory of complex rays (VTCR) when the frequency increases. This numerical strategy has indeed be mainly only used in the literature on examples where there were about 10--20 wavelengths in the characteristic length of the problem. Few works concerned its use on examples with at least two or three times more wavelengths than that, then close to the high frequency regime, which is the case here. This work tries to show how to bypass the numerical difficulties one can suffer from when using the VTCR on examples with so many wavelengths. It also studies the evolution of the variational formulation of the VTCR when the frequency increases, which leads to some interesting physical interpretations and draws relations with some high frequency vibrations strategies.From Dyson-Schwinger equations to quantum entanglementhttps://zbmath.org/1540.810302024-09-13T18:40:28.020319Z"Shojaei-Fard, Ali"https://zbmath.org/authors/?q=ai:shojaei-fard.aliSummary: We apply combinatorial Dyson-Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory \(\varPhi\) in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra \(H^{\mathrm{cut}}_{\mathrm{FG}}(\varPhi)\) and lattices of intermediate structures. Feynman diagrams in \(H_{\mathrm{FG}}(\varPhi)\) are applied to describe states in \(\varPhi\) where we build the Fisher information metric on finite dimensional linear subspaces of states in terms of homomorphism densities of Feynman graphons which are continuous functionals on the topological space \(\mathcal{S}^{\varPhi, M \subseteq [0, \infty)}_{\mathrm{graphon}}([0,1])\). We associate Hopf subalgebras of \(H_{\mathrm{FG}}(\varPhi)\) generated by quantum motions to separated regions of space-time to address some new correlations. These correlations are encoded by assigning a statistical manifold to the space of 1PI Green's functions of \(\varPhi\). These correlations are applied to build lattices of Hopf subalgebras, Lie subgroups, and Tannakian subcategories, derived from towers of combinatorial Dyson-Schwinger equations, which contribute to separated but correlated cut-distance topological regions. This lattice setting is applied to formulate a new tower of renormalization groups which encodes quantum entanglement of space-time separated particles under different energy scales.Exotic eigenvalues of shrinking metric graphshttps://zbmath.org/1540.810572024-09-13T18:40:28.020319Z"Berkolaiko, Gregory"https://zbmath.org/authors/?q=ai:berkolaiko.gregory"Colin de Verdière, Yves"https://zbmath.org/authors/?q=ai:colin-de-verdiere.yvesSummary: Eigenvalue spectrum of the Laplacian on a metric graph with arbitrary but fixed vertex conditions is investigated in the limit as the lengths of all edges decrease to zero at the same rate. It is proved that there are exactly four possible types of eigenvalue asymptotics. The number of eigenvalues of each type is expressed via the index and nullity of a form defined in terms of the vertex conditions.Kinematical waves in spacetimehttps://zbmath.org/1540.810752024-09-13T18:40:28.020319Z"Garat, Alcides"https://zbmath.org/authors/?q=ai:garat.alcidesSummary: We will prove how to create kinematical waves in spacetime. To this end we will combine the newfound technique to change locally the electromagnetic gauge in Minkowski spacetimes by using ideal solenoids and the Aharonov-Bohm effect. The local kinematical states of spacetime represented by a new kind of local tetrad will be made to oscillate according to preestablished wave equations and we will show how to produce these effects from a mathematical point of view and from a technological point of view. Kinematical waves just to mention one possible application could be used for communication.