Recent zbMATH articles in MSC 35Jhttps://zbmath.org/atom/cc/35J2023-09-22T14:21:46.120933ZUnknown authorWerkzeugFully non-linear elliptic equations on compact manifolds with a flat hyperkähler metrichttps://zbmath.org/1517.300102023-09-22T14:21:46.120933Z"Gentili, Giovanni"https://zbmath.org/authors/?q=ai:gentili.giovanni"Zhang, Jiaogen"https://zbmath.org/authors/?q=ai:zhang.jiaogenThe authors study a class of fully non-linear elliptic equations on certain compact hyperhermitian manifolds. By adapting the approach of Székelyhidi to the hypercomplex setting, they prove some a priori estimates for solutions to such equations under the assumption of existence of \(C\)-subsolutions. The main result of the paper is as follows. Let \((M, I, J, \mathrm{Keg})\) be a compact flat hyperkähler manifold, \(\Omega\) a \(q\)-real \((2,0)\)-form, and \(\underline{\varphi}\) a \(C\)-subsolution of
\[
F(A) = h,
\]
where \(h\in C^{\infty}(M, \mathbb{R})\) is given and \(F(A)=f(\lambda(A))\) is a smooth symmetric operator of the eigenvalues of \(A\). Then there exist \(\alpha \in (0,1)\) and a constant \(C>0\), depending only on \((M, I, J, K, g)\), \(\Omega\), \(h\) and \(\underline{\varphi}\), such that any \(\Gamma\)-admissible solution \(\varphi\) with \(\sup_M\varphi = 0\) satisfies the estimate
\[
|| \varphi ||_{C^{2,\alpha}} \leq C.
\]
The desired bound is obtained in two ways, by using an analogue of the Evans-Krylov theory as developed by Tosatti-Wang-Weinkove-Yang and by adapting the argument of Błocki similarly to what was done by Alesker for the treatment of the quaternionic Monge-Ampère equation.
Reviewer: Swanhild Bernstein (Freiberg)Existence of solutions to the iterative system of nonlinear two-point tempered fractional order boundary value problemshttps://zbmath.org/1517.340362023-09-22T14:21:46.120933Z"Khuddush, Mahammad"https://zbmath.org/authors/?q=ai:khuddush.mahammadSummary: In this paper we study the iterative system of nonlinear two-point tempered fractional order boundary value problems. By means of Krasnoselskii's fixed point theorem on cone, some existence results of positive solutions are obtained. The proofs are based upon the reduction of problem considered to the equivalent Fredholm integral equation of second kind. Further, we study the existence of unique solution by an application of Rus's theorem and Hyers-Ulam stability of the adderessed problem for \(\ell=1\).Radially symmetric stationary solutions for a MEMS type reaction-diffusion equation with fringing fieldhttps://zbmath.org/1517.340372023-09-22T14:21:46.120933Z"Ichida, Yu"https://zbmath.org/authors/?q=ai:ichida.yu"Sakamoto, Takashi Okuda"https://zbmath.org/authors/?q=ai:sakamoto.takashi-okudaAuthors' abstract: Radially symmetric stationary solutions for a micro-electro-mechanical system (MEMS) type reaction-diffusion equation with fringing field are considered. This equation arises in the study of the MEMS devices. This paper is devoted to the study of the existence of these solutions, information about their shape, and their asymptotic behaviour. These are studied by applying the framework that combines Poincaré type compactification, classical dynamical systems theory, and geometric methods for desingularization of vector fields called the blow-up technique.
Reviewer: Alessandro Calamai (Ancona)Elliptic regularity theory by approximation methodshttps://zbmath.org/1517.350012023-09-22T14:21:46.120933Z"Pimentel, Edgard A."https://zbmath.org/authors/?q=ai:pimentel.edgard-aPublisher's description: Presenting the basics of elliptic PDEs in connection with regularity theory, the book bridges fundamental breakthroughs -- such as the Krylov-Safonov and Evans-Krylov results, Caffarelli's regularity theory, and the counterexamples due to Nadirashvili and Vlăduţ -- and modern developments, including improved regularity for flat solutions and the partial regularity result. After presenting this general panorama, accounting for the subtleties surrounding C-viscosity and Lp-viscosity solutions, the book examines important models through approximation methods. The analysis continues with the asymptotic approach, based on the recession operator. After that, approximation techniques produce a regularity theory for the Isaacs equation, in Sobolev and Hölder spaces. Although the Isaacs operator lacks convexity, approximation methods are capable of producing Hölder continuity for the Hessian of the solutions by connecting the problem with a Bellman equation. To complete the book, degenerate models are studied and their optimal regularity is described.Controlling monotonicity of nonlinear operatorshttps://zbmath.org/1517.350082023-09-22T14:21:46.120933Z"Borowski, Michał"https://zbmath.org/authors/?q=ai:borowski.michal"Chlebicka, Iwona"https://zbmath.org/authors/?q=ai:chlebicka.iwonaSummary: Controlling the monotonicity and growth of Leray-Lions' operators including the \(p\)-Laplacian plays a fundamental role in the theory of existence and regularity of solutions to second order nonlinear PDEs. We collect, correct, and supply known estimates including the discussion on the constants. Moreover, we provide a comprehensive treatment of related results for operators with Orlicz growth. We pay special attention to exposition of the proofs and the use of elementary arguments.On removable singularities for solutions of Neumann problem for elliptic equations involving variable exponenthttps://zbmath.org/1517.350092023-09-22T14:21:46.120933Z"Apaza, Juan Alcon"https://zbmath.org/authors/?q=ai:alcon-apaza.juanSummary: We study the removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case where the singular set is compact, and give sufficient conditions for removability of this singularity for equations in the variable exponent Sobolev space \(W^{1,p(\cdot)}(\Omega)\).Phase separating solutions for two component systems in general planar domainshttps://zbmath.org/1517.350182023-09-22T14:21:46.120933Z"Kowalczyk, Michał"https://zbmath.org/authors/?q=ai:kowalczyk.michal"Pistoia, Angela"https://zbmath.org/authors/?q=ai:pistoia.angela"Vaira, Giusi"https://zbmath.org/authors/?q=ai:vaira.giusiThis paper is concerned with the construction of phase separating solutions for the following the system:
\[
\begin{aligned}
&-\Delta u_1 = f (u_1, x) - \beta u_1u^2_2 \text{ in }\Omega,\\
&-\Delta u_2 = f (u_2, x) -\beta u_2u^2_1 \text{ in } \Omega,\\
& \hspace{2cm} u_1|_{\partial \Omega} = u_2 |_{\partial \Omega}= 0,
\end{aligned}\tag{1}
\]
as the parameter \(\beta\) is large enough. Here \(\Omega \subset \mathbb{R}^2\) is a bounded with smooth boundary. If the function \(f \in C^{3,\gamma} (\mathbb{R} \times \Omega)\) with \(\gamma \in (0,1)\), and \(w \in H^s(\Omega) \ ( s > 11/2 )\) is a non-degenerate solution (see the definition in the paper) to the Dirichlet problem
\[
- \Delta w = f (w, x) \text{ in } \Omega, \ w|_{\partial \Omega} = 0,
\]
the authors prove that the system (1) has a solution \((u_1, u_2)\) such that, as \(\beta\rightarrow +\infty\),
\[
\|w_i - u_i \|_{C^\alpha (\Omega) }= \mathcal{O} (\beta^{-(1\alpha)/4}) \text{ with } \alpha \in [0,1),
\]
and, for some \(\alpha \in (0,1/2)\),
\[
\|w_i - u_i \|_{C^{2,\alpha}(K)} = \mathcal{O} (\beta^{-1/4}),
\]
over the compacts \(K \subset \Omega\setminus \Gamma\) and \(\Gamma = \{x\in \Omega: w(x) = 0\}.\)
Reviewer: Huansong Zhou (Wuhan)Stochastic homogenization: a short proof of the annealed Calderón-Zygmund estimatehttps://zbmath.org/1517.350202023-09-22T14:21:46.120933Z"Josien, Marc"https://zbmath.org/authors/?q=ai:josien.marcSummary: A building block for many field theories in continuum physics are second-order elliptic operators in divergence form, as given through a coefficient field which may be assimilated to a metric tensor field on \(\mathbb{R}^d\). The mapping properties of these linear operators are a crucial ingredient for analysis. In this paper, we focus on Calderón-Zygmund estimates, that is, on the boundedness of the corresponding Helmholtz projection in \(\mathrm{L}^p (\mathbb{R}^d)\)-spaces. Even when the coefficient field is uniformly smooth, this estimate may fail for \(p\) not close to 2. We seek an intrinsic criterion on the validity of the Calderón-Zygmund estimate in the whole range of \(p\in (1,\infty)\); intrinsic in the sense that it is formulated in terms of the scalar and vector potentials of the harmonic coordinates. We seek genericity in form of a statistical statement, and thus consider general ensembles of coefficient fields. Our criterion comes in form of finite stochastic moments for the potentials, or rather their corrections from being affine. In line with this, the Calderón-Zygmund estimates we obtain are \textit{annealed} as opposed to \textit{quenched}, meaning that there is an inner norm in form of a stochastic moment next to the (outer) \(\mathrm{L}^p\)-norm in space. This result grows out of recent progress in quantitative stochastic homogenization; it is ultimately inspired by the classical large-scale regularity theory of \textit{M. Avellaneda} and \textit{F.-H. Lin} [Commun. Pure Appl. Math. 40, No. 6, 803--847 (1987; Zbl 0632.35018)]. More specifically, we provide an easier version of the proof given by us in [with \textit{F. Otto}, J. Funct. Anal. 283, No. 7, Article ID 109594, 74 p. (2022; Zbl 1494.35022)], albeit under stronger assumptions. Annealed Calderon-Zygmund estimates were first established in [\textit{M. Duerinckx} and \textit{F. Otto}, Stoch. Partial Differ. Equ., Anal. Comput. 8, No. 3, 625--692 (2020; Zbl 1456.35017)], and are a very convenient tool for error estimates in stochastic homogenization.High contrasting diffusion in Heisenberg group: homogenization of optimal control via unfoldinghttps://zbmath.org/1517.350212023-09-22T14:21:46.120933Z"Nandakumaran, A. K."https://zbmath.org/authors/?q=ai:nandakumaran.akambadath-keerthiyil"Sufian, Abu"https://zbmath.org/authors/?q=ai:sufian.abuSummary: The periodic unfolding method is one of the latest tools for studying multiscale problems like homogenization after the development of multiscale convergence in the 1990s. It provides a good understanding of various microscales involved in the problem, which can be conveniently and easily applied to get the asymptotic limit. In this article, we develop \textit{the periodic unfolding} for the Heisenberg group, which has a noncommutative group structure. The concept of greatest integer part and fractional part for the Heisenberg group has been introduced corresponding to the periodic cell. Analogous to the Euclidean unfolding operator, we prove the integral equality, \(L^2\)-weak compactness, unfolding gradient convergence, and other related properties. Moreover, we have the adjoint operator for the unfolding operator, which can be recognized as an average operator. As an application of the unfolding operator, we have homogenized the standard elliptic PDE with oscillating coefficients. We have also considered an optimal control problem with the state equation having high contrasting diffusivity coefficients. The high contrasting coefficients are an added difficulty in the analysis. Moreover, we have characterized the interior periodic optimal control in terms of the unfolding operator, which helps us to analyze the asymptotic behavior.Maximum principles for Laplacian and fractional Laplacian with critical integrabilityhttps://zbmath.org/1517.350762023-09-22T14:21:46.120933Z"Li, Congming"https://zbmath.org/authors/?q=ai:li.congming"Lü, Yingshu"https://zbmath.org/authors/?q=ai:lu.yingshuIt is well known that, given an open connected set \(\Omega\) in \(\mathbb{R}^n\) (\(n\geq 3\)) and a function \(c:\Omega \rightarrow \mathbb{R}\), the strong maximum principle for the Schrödinger operator \(-\Delta +c(x)\) holds when \(c\in L^p(\Omega)\), for some \(p>\frac{n}{2}\), and fails if \(p<\frac{n}{2}\).
The first main result of this paper states that the critical integrability condition \(c\in L^{\frac{n}{2}}(\Omega)\) is not sufficient for the validity of the strong maximum principle for the operator \(-\Delta +c(x)\), even if \(\|c\|_{L^{\frac{n}{2}}(\Omega)}>0\) is small.
Next, the validity of the maximum principle is investigated for the more general operator \(-\Delta +\vec{b}(x)\nabla +c(x)\) in a smooth bounded open domain \(\Omega\), where, in this case, the critical integrability condition for the coefficient \(\vec{b}(x)\) is \(\vec{b} \in L^n(\Omega)\).
As a result, the authors show that the maximum principle holds when \(\vec{b} \in L^n(\Omega)\), \(c\in L^{\frac{n}{2}}(\Omega)\), and \(\|\vec{b}\|_{L^n(\Omega)}+\|c^-\|_{L^{\frac{n}{2}}(\Omega)}\) is small.
As a corollary, the validity of a strong maximum principle for the operator \(-\Delta +\vec{b}(x)\nabla\), when \(\vec{b} \in L^n(\Omega)\) and \(\|\vec{b}\|_{L^n(\Omega)}\) is small, is derived.
Finally, the authors investigate the maximum principle and the strong maximum principle for operators involving the fractional Laplacian \((-\Delta)^s\), \(s\in (0,1)\). In particular, they improve a known result on the validity of maximum principle for the distributional solutions \(u\) of \((-\Delta)^su\geq 0\) in \(\Omega\) by removing the lower semicontinuous assumption on \(u\), and (partially) extend the above results to the operator \((-\Delta)^s +\vec{b}(x)\nabla +c(x)\).
Reviewer: Giovanni Anello (Messina)On smoothness of solution of a class of \(p\)-harmonic type equationshttps://zbmath.org/1517.350792023-09-22T14:21:46.120933Z"Najafov, Alik M."https://zbmath.org/authors/?q=ai:najafov.alik-malik-oglu"Alekberli, Sain T."https://zbmath.org/authors/?q=ai:alekberli.sain-tThe authors study existence and uniqueness of a Dirichlet problem related to \(p\)-harmonic type equations, in grand Sobolev spaces \(W^1_{p)}(G) \), where \(G\) is a bounded domain in \(R^n\) with a non-smooth boundary.
They apply the embedding theorems in grand Sobolev-Morrey spaces for studying smoothness of solutions.
Let us point out that the results are proved stating that the solution belongs to the Hölder class inside the domain and has a zero boundary Dirichlet condition up to bounds.
Reviewer: Maria Alessandra Ragusa (Catania)An observation on the Dirichlet problem at infinity in Riemannian coneshttps://zbmath.org/1517.351002023-09-22T14:21:46.120933Z"Cortissoz, Jean C."https://zbmath.org/authors/?q=ai:cortissoz.jean-cSummary: In this short paper, we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below). This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied to smooth Riemannian manifolds with a special type of metrics, which generalize the class of metrics with rotational symmetry, we obtain generalizations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODEs.Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroupshttps://zbmath.org/1517.351012023-09-22T14:21:46.120933Z"De Nápoli, Pablo Luis"https://zbmath.org/authors/?q=ai:de-napoli.pablo-luis"Stinga, Pablo Raúl"https://zbmath.org/authors/?q=ai:stinga.pablo-raulSummary: In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by \textit{S. Minakshisundaram}'s ideas [J. Indian Math. Soc., New Ser. 13, 41--48 (1949; Zbl 0033.11605)], we find a precise pointwise description of \((-\Delta _{\mathbb{S}^{n-1}})^su(x)\) in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, additional pointwise integro-differential formulas are obtained. Finally, we prove a characterization with a local extension problem and the interior Harnack inequality.
For the entire collection see [Zbl 1416.35007].Liouville type theorems for positive harmonic functions on the unit ball with a nonlinear boundary conditionhttps://zbmath.org/1517.351022023-09-22T14:21:46.120933Z"Lin, Daowen"https://zbmath.org/authors/?q=ai:lin.daowen"Ou, Qianzhong"https://zbmath.org/authors/?q=ai:ou.qianzhongThe paper under review deals with Liouville type theorems for positive harmonic functions on the unit ball. More precisely, the authors consider the problem
\begin{align*}
\Delta u&=0 \quad \text{in }\mathbb{B}^n \, ,\\
u_\nu+\lambda u&=u^q\quad \text{on }\mathbb{S}^{n-1}\, .
\end{align*}
They prove that if \(u \in C^{\infty}(\overline{\mathbb{B}^n})\) is a positive solution of the problem above when \(n \geq 9\) and \(1<q\leq 1+\frac{2}{3n-5}\), then there is a \(\lambda_0 \in (0,\frac{1}{q-1})\) (which depends on \(n\) and on \(q\)) such that \(u\) is constant if \(\lambda \in (0,\lambda_0)\). The proof is based on integral identities and Sobolev inequalities.
Reviewer: Paolo Musolino (Padova)Mixed boundary value problems for the Helmholtz equationhttps://zbmath.org/1517.351032023-09-22T14:21:46.120933Z"Natroshvili, David"https://zbmath.org/authors/?q=ai:natroshvili.david"Tsertsvadze, Tornike"https://zbmath.org/authors/?q=ai:tsertsvadze.tornikeSummary: We consider a new approach to investigating the mixed boundary value problem (BVP) for the Helmholtz equation in the case of a three-dimensional unbounded domain \(\Omega^-\subset\mathbb{R}^3\) with compact boundary surface \(S=\partial\,\Omega^-\), which is divided into two disjoint parts, \(S_D\) and \(S_N\), where the Dirichlet and Neumann type boundary conditions are prescribed respectively. Our approach is based on the classical potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP to a system of integral equations containing neither extensions of the Dirichlet or Neumann data nor a Steklov-Poincaré type operator involving the inverse of the single layer boundary integral operator, which is not available explicitly for arbitrary boundary surfaces. The right-hand sides of the resulting boundary integral equations system are functions coinciding with the given Dirichlet and Neumann data of the problem under consideration. We show that the corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate \(L_2\)-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the class of functions belonging to the Sobolev space \(W_{2,\mathrm{loc}}^1(\Omega^-)\) and satisfying the Sommerfeld radiation conditions. We also show that the pseudodifferential matrix operator thus obtained is invertible in the \(L_p\)-based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses the \(C^\alpha\)-Hölder continuity property in the closed domain \(\overline\Omega^-\) with \(\alpha=\frac12-\varepsilon\), where \(\varepsilon > 0\) is an arbitrarily small number.New type of solutions for the nonlinear Schrödinger equation in \(\mathbb{R}^N\)https://zbmath.org/1517.351042023-09-22T14:21:46.120933Z"Duan, Lipeng"https://zbmath.org/authors/?q=ai:duan.lipeng"Musso, Monica"https://zbmath.org/authors/?q=ai:musso.monicaSolutions \(u \in H^1(\mathbb{R}^N)\) of the nonlinear Schrödinger equation
\[
-\Delta u + V(|y|)u = u^p, \quad u>0\tag{1}
\]
are constructed when \(1<p<\frac{N+2}{N-2}\), \(N \ge 3\), and the radial potential \(V\) is positive, bounded and \[ V(|y|) =V_0 + \frac{a}{|y|^m} + O\left(\frac{1}{|y|^{m+\sigma}}\right) \text{ as }\: |y| \to \infty \] for some constants \(V_0, a,\sigma>0 \) and \(m> \max\{\frac{4}{p-1},2\}\). More precisely, let \(U\) be the radial solution of (1) when \(V =0\) and so that \(U(y) \to 0\) as \(|y|\to 0\). Define \(W_{r,h}(y)= \sum_{j=1}^{2k}U(y-x_j)\) where the \(2k\) points \(x_j\) are symmetrically chosen on the sphere \(y_1^2+y_2^2+y_3^2=r^2\) intersected with the two 2D planes \(y_3=\pm rh\), \(y_j=0\), \(4\le j \le N\). The parameters \(r,h>0\) are chosen in a small range dependent on \(k\). The main result is: For all \(k\) large enough there is a solution \(u_k\) of (1) of the form \(u_k = W_{r_k,h_k}+\omega_k\) where \(\omega_k\in H^1(\mathbb{R}^N)\) has certain symmetry properties and \[ \int_{\mathbb{R}^N}|\nabla \omega_k|^2+V|\omega_k|^2 \to 0 \text{ as } k \to \infty. \] This result is related to that of \textit{J. Wei} and \textit{S. Yan} [Calc. Var. Partial Differ. Equ. 37, No. 3--4, 423--439 (2010; Zbl 1189.35106)]. The latter result is roughly equivalent to taking \(h=0\) above.
Reviewer: Denis A. White (Toledo)On critical double phase Kirchhoff problems with singular nonlinearityhttps://zbmath.org/1517.351052023-09-22T14:21:46.120933Z"Arora, Rakesh"https://zbmath.org/authors/?q=ai:arora.rakesh"Fiscella, Alessio"https://zbmath.org/authors/?q=ai:fiscella.alessio"Mukherjee, Tuhina"https://zbmath.org/authors/?q=ai:mukherjee.tuhina"Winkert, Patrick"https://zbmath.org/authors/?q=ai:winkert.patrickLet $\Omega\subset\mathbb{R}^N$ be a bounded domain with Lipschitz boundary $\partial\Omega$. This work combines the effects of a nonlocal Kirchhoff coefficient $m:[0,+\infty)\to [0,+\infty)$ given as
\[
m(t)=a_0+b_0t^{\theta-1}\text{ for all }t\geq0,\text{ with }a_0\geq 0, b_0>0\text{ and }\theta \in [1,p^\ast/q)
\]
($p^\ast=Np/(N-p)$ is the critical Sobolev exponent to $p$, $1<p<N$ and $p<q<p^\ast$), and a double phase operator
\[
\operatorname{div}(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u)\text{ for all }u\in W_0^{1,\mathcal{H}}(\Omega)
\]
($W_0^{1,\mathcal{H}}(\Omega)$ is the homogeneous Musielak-Orlicz Sobolev space) with a parametric singular term $\lambda u^{-\gamma}$ ($\lambda>0$ and $0<\gamma<1$) and a critical term $u^{p^\ast-1}$. The proof of the main result is based on a minimization argument on the Nehari manifold. Under appropriate hypotheses on the data, the authors obtain the existence of at least a weak solution $u_\lambda \in W_0^{1,\mathcal{H}}(\Omega)$ to the Dirichlet problem for appropriate values of $\lambda$.
Reviewer: Calogero Vetro (Palermo)The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifoldshttps://zbmath.org/1517.351062023-09-22T14:21:46.120933Z"Guan, Bo"https://zbmath.org/authors/?q=ai:guan.bo|guan.bo.1The author considers the Dirichlet problem for a class of fully nonlinear elliptic equations in Euclidean space and on Riemannian manifolds.
More precisely, let \((\overline{M}^n,g)\) be a compact Riemannian manifold of dimension \(n\ge 2\) with smooth boundary \(\partial M\). Denote with \(M\) the interior of \(\overline{M}\).
The Dirichlet problem for fully nonlinear elliptic equations considered is
\[
f(\lambda(\nabla^2u+\chi))=\psi \text{ in } \overline{M}, \ \ \ u=\varphi \text{ on } \partial M,
\]
where it is assumed \(f\) is a smooth, symmetric function of \(n\) variables, \(\chi\) is a smooth \((0,2)\) tensor on \(M\), \(u\in C^2(M)\) and \(\lambda(\nabla^2u+\chi)=(\lambda_1,\ldots,\lambda_n)\) are the eigenvalues of \(\nabla^2 u+\chi\) with respect to the metric \(g\). Assume also \(\psi\in C^{\infty}(\overline{M})\) and \(\varphi\in C^{\infty}(\partial M)\).
Using structure conditions from [\textit{L. Caffarelli} et al., Acta Math. 155, 261--301 (1985; Zbl 0654.35031)], the author proves uniqueness of a smooth solution to the Dirichlet problem, provided there exists an admissible subsolution, thus extending the existence result of Caffarelli et al. [loc. cit.] and of \textit{N. S. Trudinger} [Acta Math. 175, No. 2, 151--164 (1995; Zbl 0887.35061)]. The theorem proved here is optimal in the sense that if any of the structure conditions are removed, it no longer holds.
Reviewer: Mariana Vega Smit (Bellingham)A two-way coupled model of visco-thermo-acoustic effects in photoacoustic trace gas sensorshttps://zbmath.org/1517.351072023-09-22T14:21:46.120933Z"Mozumder, Ali"https://zbmath.org/authors/?q=ai:mozumder.ali"Safin, Artur"https://zbmath.org/authors/?q=ai:safin.artur"Minkoff, Susan E."https://zbmath.org/authors/?q=ai:minkoff.susan-e"Zweck, John"https://zbmath.org/authors/?q=ai:zweck.john-wSummary: We introduce the first two-way coupled model for the thermo-viscous damping of a mechanical structure (such as quartz tuning fork) that is forced by the weak acoustic and thermal waves generated when a laser source periodically interacts with a trace gas. The model is based on a Helmholtz system of thermo-visco-acoustic equations in the fluid, together with a system of equations for the temperature and the displacement of the structure. These two subsystems are coupled across the fluid-structure interface via several conditions. With this model, the user specifies the geometry of the structure and the viscous and thermal parameters of the fluid, and the model outputs an effective damping parameter and a signal strength that is proportional to the concentration of the trace gas. This new model is a significant improvement over existing one-way coupled models in which damping effects are incorporated via a priori laboratory measurements. Analytical solutions derived for an annular structure show reasonable agreement between the one-way and two-way coupled models at higher ambient pressures. However, at low ambient pressure the one-way coupled model does not adequately capture thermo-viscous effects.Realisations of elliptic operators on compact manifolds with boundaryhttps://zbmath.org/1517.351082023-09-22T14:21:46.120933Z"Bandara, Lashi"https://zbmath.org/authors/?q=ai:bandara.lashi"Goffeng, Magnus"https://zbmath.org/authors/?q=ai:goffeng.magnus"Saratchandran, Hemanth"https://zbmath.org/authors/?q=ai:saratchandran.hemanthThe authors analyze realizations of elliptic differential operators on compact manifolds with boundary. In contrast to the traditional approach, which sees a realization as imposing equations that solutions must satisfy on the boundary, in this paper the authors use the Bär-Ballmann approach to first order elliptic operators. That means studying realizations of an elliptic operator through the quotient between the domains of maximal and minimal realization. This paper therefore proposes a perspective inspired by \textit{C. Bär} and \textit{W. Ballmann} [Surv. Differ. Geom. 17, 1--78 (2012; Zbl 1331.58022)], giving an overview of the theory for elliptic boundary problems from the perspective of the Cauchy data space. The Cauchy data space is studied using the Calderon projection from [\textit{R. T. Seeley}, Am. J. Math. 88, 781--809 (1966; Zbl 0178.17601)].
Reviewer: Mariana Vega Smit (Bellingham)Uniqueness for the nonlocal Liouville equation in \(\mathbb{R}\)https://zbmath.org/1517.351092023-09-22T14:21:46.120933Z"Ahrend, Maria"https://zbmath.org/authors/?q=ai:ahrend.maria"Lenzmann, Enno"https://zbmath.org/authors/?q=ai:lenzmann.ennoAuthors' abstract: We prove uniqueness of solutions for the nonlocal Liouville equation
\[
(-\Delta)^{1/2}w=Ke^w\text{ in } \mathbb{R}
\]
with finite total \(Q\)-curvature \(\int_{\mathbb{R}}K e^wdx<+\infty\). Here the prescribed \(Q\)-curvature function \(K=K(|x|)>0\) is assumed to be a positive, symmetric-decreasing function satisfying suitable regularity and decay bounds. In particular, we obtain uniqueness of solutions in the Gaussian case with \(K(x)=\exp(-x^2)\). Our uniqueness proof exploits a connection of the nonlocal Liouville equation to ground state solitons for Calogero-Moser derivative NLS, which is a completely integrable PDE recently studied by \textit{P. Gérard} and the second author [``The Calogero-Moser Derivative Nonlinear Schrödinger Equation'', Preprint, \url{arXiv;2298.04105}].
Reviewer: Dian K. Palagachev (Bari)PWB-method and Wiener criterion for boundary regularity under generalized Orlicz growthhttps://zbmath.org/1517.351102023-09-22T14:21:46.120933Z"Benyaiche, Allami"https://zbmath.org/authors/?q=ai:benyaiche.allami"Khlifi, Ismail"https://zbmath.org/authors/?q=ai:khlifi.ismailSummary: Perron's method and Wiener's criterion, the Dirichlet problem associated with the Laplacian equation was entirely solved. Since then, these ideas have been used for more general equations. So, in this paper, we extend these methods to study the regularity of boundary points of bounded domain concerning the Dirichlet problem associated with quasilinear equations under Musielak-Orlicz growth.Critical perturbations for second-order elliptic operators. I: Square function bounds for layer potentialshttps://zbmath.org/1517.351112023-09-22T14:21:46.120933Z"Bortz, Simon"https://zbmath.org/authors/?q=ai:bortz.simon"Hofmann, Steve"https://zbmath.org/authors/?q=ai:hofmann.steve"Luna García, José Luis"https://zbmath.org/authors/?q=ai:luna-garcia.jose-luis"Mayboroda, Svitlana"https://zbmath.org/authors/?q=ai:mayboroda.svitlana"Poggi, Bruno"https://zbmath.org/authors/?q=ai:poggi.bruno.1The main goal of the authors is the study of the \(L^2\) Dirichlet, Neumann and regularity problems for critical perturbations of second-order divergence form equations by lower-order terms. This is the first of two papers in which the authors establish square function bounds for an abstract class of layer potential operators, and obtain uniform slice bounds for solutions (in terms of the square function).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Geometric aspects of shape optimizationhttps://zbmath.org/1517.351122023-09-22T14:21:46.120933Z"Plotnikov, Pavel I."https://zbmath.org/authors/?q=ai:plotnikov.pavel-i"Sokolowski, Jan"https://zbmath.org/authors/?q=ai:sokolowski.janSummary: We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model problem. To our best knowledge it is the first result of convergence in shape optimization. The complete proofs of some results are presented in report [the authors, ``Gradient flow for Kohn-Vogelius functional'', Siberian Electron. Math. Rep., to appear, \url{https://hal.science/hal-03896975}].Singular anisotropic problems with competition phenomenahttps://zbmath.org/1517.351132023-09-22T14:21:46.120933Z"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.francescaGiven a bounded domain \(\Omega\subseteq \mathbb{R}^N\) with a \(C^2\)-boundary \(\partial\Omega\), the authors consider a singular anisotropic Dirichlet problem of the form
\[
\begin{cases} -\Delta_{p(z)} u(z)-\Delta_{q(z)}u(z)=\lambda \left[u(z)^{-\eta(z)}+u(z)^{\tau(z)-1}\right]+f(z,u(z))&\text{in }\Omega,\\
u=0 &\text{on }\partial \Omega, \end{cases}
\]
where \(\lambda>0\),
\begin{align*}
&1<\tau_-\leq\tau_+<q_-\leq q_+<p_-\leq p_+,\\
& 0<\eta_-\leq\eta_+<1,
\end{align*}
and \(f\colon \Omega\times\mathbb{R}\to\mathbb{R}\) is a Carathéodory function which exhibits \((p_+-1)\)-superlinear growth at infinity. Using variational tools together with truncation and comparison techniques, the authors prove a global existence and multiplicity theorem.
Reviewer: Patrick Winkert (Berlin)Local estimates for conformal \(Q\)-curvature equationshttps://zbmath.org/1517.351142023-09-22T14:21:46.120933Z"Jin, Tianling"https://zbmath.org/authors/?q=ai:jin.tianling"Yang, Hui"https://zbmath.org/authors/?q=ai:yang.hui.4The main result of the paper is a local upper bound for positive singular solutions \(u\) the (higher-order) prescribed \(Q\)-curvature equation
\[
(-\Delta)^m u = K(x) u^\frac{n+2m}{n-2m} \quad \text{ on } \Omega \setminus \Lambda,
\]
where \(m \in [1, n/2)\) is an integer, \(K(x)\) is the prescribed \(Q\)-curvature and \(\Omega \subset\mathbb R^n\) is an open set. The singular set \(\Lambda \subset \mathbb R^n\), which is such that \(u(x) \to \infty\) as \(x \to\Lambda\), is assumed to be of dimension at most \((n-2m)/2\). Then, under a certain flatness condition for critical points of \(K\) on \(\Lambda\), the bound \(u(x) \leq C \text{dist}(x,\Lambda)^{-\frac{n-2m}{2}}\) for \(x\) close to \(\Lambda\) is shown.
The proof proceeds by rewriting the equation for \(u\) as the integral equation
\[
u(x) = \int_{B_2} \frac{K(y) u^\frac{n+2m}{n-2m}(y)}{|x-y|^{n-2m}} \, dy + h(x) \qquad \text{ on } B_2 \setminus \Sigma,
\]
with \(h \in C^1(B_2)\), and deriving the corresponding estimates for solutions of this equation. The analysis of the integral equation actually covers every real number \(m \in (0, \frac{n}{2})\).
Reviewer: Tobias König (Frankfurt am Main)Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearityhttps://zbmath.org/1517.351152023-09-22T14:21:46.120933Z"Ma, Shiwang"https://zbmath.org/authors/?q=ai:ma.shiwang"Moroz, Vitaly"https://zbmath.org/authors/?q=ai:moroz.vitalyThe topic of this article is the asymptotic behavior of positive radially symmetric ground state solutions to the nonlinear Schrödinger equation with double power nonlinearity
\[
-\Delta u_\lambda + u_\lambda = u_\lambda^{2^* - 1} + \lambda u_\lambda^{q-1} \quad \text{ in } \mathbb R^N, \quad \lambda > 0,
\]
as \(\lambda \to 0\), where \(N \geq 3\), \(2^* = \frac{2N}{N-2}\) is the critical Sobolev exponent and \(q \in (2, 2^*)\) is subcritical.
In the limit \(\lambda \to 0\), for \(\mu_\lambda = u_\lambda(0)\), the rescaling \(v_\lambda(x) = \mu_\lambda^{-1} u_\lambda( \mu_\lambda^\frac{2}{N-2} x)\) converges to a solution \(U\) of the limit equation \(-\Delta U = U^{2^* - 1}\). In this paper, the authors derive the explicit growth rate of \(\mu_\lambda\) in terms of \(\lambda\). This growth rate is sensitive to the dimension and takes a different expression if \(N \geq 5\), \(N = 4\) and \(N = 3\), respectively. Similar dimension-dependent asymptotics rates are obtained for \(\|u_\lambda\|_2\) and \(\|u_\lambda\|_q\).
Under certain additional assumptions on \(N\) and \(q\), the authors also deduce from this the blow-up asymptotics for ground states of the analogous \(L^2\)-mass-constrained variational problem with double power nonlinearity, where the parameter \(\lambda \to 0\) is replaced by the constraint \(\rho = \|u\|_2 \to 0\).
Reviewer: Tobias König (Frankfurt am Main)A note on second derivative estimates for Monge-Ampère-type equationshttps://zbmath.org/1517.351162023-09-22T14:21:46.120933Z"Trudinger, Neil S."https://zbmath.org/authors/?q=ai:trudinger.neil-sThe author studies Pogorelov-type interior and global second derivative estimates for solutions of Monge-Ampére-type equations, returning to the works of \textit{F. Jiang} and the author [Bull. Math. Sci. 4, No. 3, 407--431 (2014; Zbl 1307.90023)], \textit{J. Liu} and the author [Discrete Contin. Dyn. Syst. 28, No. 3, 1121--1135 (2010; Zbl 1387.35317)] and the author [Math. Eng. (Springfield) 3, No. 6, Paper No. 48, 17 p. (2021; Zbl 1512.90039)]. Here, the key ingredients are an extension of the Pogorelov-type estimate from [\textit{J. Liu} and the author, Discrete Contin. Dyn. Syst. 28, No. 3, 1121--1135 (2010; Zbl 1387.35317)], a result from [the author, Math. Eng. (Springfield) 3, No. 6, Paper No. 48, 17 p. (2021; Zbl 1512.90039)], and a strict convexity result of \textit{N. Guillen} and \textit{J. Kitagawa} [Commun. Pure Appl. Math. 70, No. 6, 1146--1220 (2017; Zbl 1375.35234)]. The results can also be further improved using the work of \textit{C. Rankin} [Calc. Var. Partial Differ. Equ. 60, No. 6, Paper No. 221, 14 p. (2021; Zbl 1479.35504)].
More precisely, let \(\mathcal{U}\subset\mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n\) be a domain, \(A\) be an \(n\times n\) symmetric matrix-valued function defined on \(\mathcal{U}\), and \(B\) a scalar function defined on \(\mathcal{U}\). Assume \(A, B\in C^2\), \(B>0\) and \(A\) to be regular with respect to the gradient variables. Denoting points in \(\mathcal{U}\) by \((x,z,p)\), assume \(\mathcal{U}\) is convex in \(p\). Let \(\Omega\subset\mathbb{R}^n\) be a bounded domain, \(u\in C^2(\Omega)\) with \(J_1[u](\Omega)\subset \mathcal{U}\).
The equations considered are of the form
\[
\mathrm{det}[D^2u-A(\cdot, u, Du)]=B(\cdot, u, Du).
\]
The author first addresses Pogorelov estimates and applications to second derivative bounds for solutions to boundary value problems for generated Jacobian equations. Applying this to the existence of globally smooth classical solutions, the author is able to remove the monotonicity conditions on the matrix function \(A\).
Reviewer: Mariana Vega Smit (Bellingham)Steady states of a diffusive predator-prey model with prey-taxis and fear effecthttps://zbmath.org/1517.351202023-09-22T14:21:46.120933Z"Cao, Jianzhi"https://zbmath.org/authors/?q=ai:cao.jianzhi"Li, Fang"https://zbmath.org/authors/?q=ai:li.fang|li.fang.5|li.fang.4|li.fang.1|li.fang.2|li.fang.7"Hao, Pengmiao"https://zbmath.org/authors/?q=ai:hao.pengmiaoSummary: In this paper, a diffusive predator-prey system with a prey-taxis response subject to Neumann boundary conditions is considered. The stability, the Hopf bifurcation, the existence of nonconstant steady states, and the stability of the bifurcation solutions of the system are analyzed. It is proved that a high level of prey-taxis can stabilize the system, the stability of the positive equilibrium is changed when \(\chi\) crosses \(\chi_0\), and the Hopf bifurcation occurs for the small \(s\). The system admits nonconstant positive solutions around \((\overline{u},\overline{v},\chi_i )\), the stability of bifurcating solutions are controlled by \(\int_{\Omega}\Phi_i^3\text{d}x\) and \(\int_{\Omega}\Phi_i^4\text{d}x\). Finally, numerical simulation results are carried out to verify the theoretical findings.Surface concentration of transmission eigenfunctionshttps://zbmath.org/1517.351412023-09-22T14:21:46.120933Z"Chow, Yat Tin"https://zbmath.org/authors/?q=ai:chow.yat-tin"Deng, Youjun"https://zbmath.org/authors/?q=ai:deng.youjun"Liu, Hongyu"https://zbmath.org/authors/?q=ai:liu.hongyu"Sunkula, Mahesh"https://zbmath.org/authors/?q=ai:sunkula.maheshThe authors of this article consider an eigenvalue problem coming from acoustic scattering theory. Specifically, they deal with the interior transmission eigenvalue problem which is a system of two coupled partial differential equations along with transmission boundary conditions. This leads to a non-elliptic and non-selfadjoint spectral problem. The eigensolutions to this problem are closely related to non-scattering waves and hence to invisibility. Another aspect is that reconstruction algorithms like Kirsch's factorization method or the (generalized) sampling method are not justified for such wave numbers that are the corresponding eigenvalues, the so called interior transmission eigenvalues. In a previous paper, some of the authors numerically observed that the eigenfunctions concentrate on the boundary of the given domain and close to it. This is fundamentally different to classical eigenvalue problems, for example the eigenfunctions of the negative Laplacian with Dirichlet or Neumann boundary condition. In this article, the authors theoretically verify in detail this interesting behavior for the transmission eigenfunctions by reformulating the interior transmission problem using boundary integral operators combined with the generalized Weyl's law and certain novel ergodic properties concentrating specifically on the three dimensional case.
Reviewer: Andreas Kleefeld (Jülich)Non-monotonicity of the first eigenvalue for the 3D magnetic Robin Laplacianhttps://zbmath.org/1517.351432023-09-22T14:21:46.120933Z"Miranda, Germán"https://zbmath.org/authors/?q=ai:miranda.germanSummary: Previous works provided several counterexamples to monotonicity of the lowest eigenvalue for the magnetic Laplacian in the two-dimensional case. However, the three-dimensional case is less studied. We use the results obtained by Helffer, Kachmar, and Raymond to provide one of the first counterexamples in 3D. Considering the magnetic Robin Laplacian on the unit ball with a constant magnetic field, we show the non-monotonicity of the lowest eigenvalue asymptotics when the Robin parameter tends to \(+\infty\).Maximization of Neumann eigenvalueshttps://zbmath.org/1517.351442023-09-22T14:21:46.120933Z"Bucur, Dorin"https://zbmath.org/authors/?q=ai:bucur.dorin"Martinet, Eloi"https://zbmath.org/authors/?q=ai:martinet.eloi"Oudet, Edouard"https://zbmath.org/authors/?q=ai:oudet.edouardSummary: This paper is motivated by the maximization of the \(k\)-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of \({{\mathbb{R}}}^N\) with prescribed measure. We relax the problem to the class of (possibly degenerate) densities in \(\mathbb{R}^N\) with prescribed mass and prove the existence of an optimal density. For \(k=1,2\), the two problems are equivalent and the maximizers are known to be one and two equal balls, respectively. For \(k \ge 3\) this question remains open, except in one dimension of the space, where we prove that the maximal densities correspond to a union of \(k\) equal segments. This result provides sharp upper bounds for Sturm-Liouville eigenvalues and proves the validity of the Pólya conjecture in the class of densities in \(\mathbb{R} \). Based on the relaxed formulation, we provide numerical approximations of optimal densities for \(k=1, \dots , 8\) in \(\mathbb{R}^2\).Monotonicity of eigenvalues of the fractional \(p\)-Laplacian with singular weightshttps://zbmath.org/1517.351462023-09-22T14:21:46.120933Z"Iannizzotto, Antonio"https://zbmath.org/authors/?q=ai:iannizzotto.antonioIn this paper, the authors considered the following nonlinear eigenvalue problem
\[
\left\{ \begin{array}{ll} (-\Delta)_p^s u = \lambda m(x)|u|^{p-2}u, \quad & x\in \Omega,\\
u=0 , & x\in {\Omega}^c, \end{array} \right.
\]
where \(\Omega \subset \mathbb{R}^N\) (\(N\ge 2\)) is a bounded open domain, \(p>1\), \(s\in (0,1)\) such that \(ps <N\) and \(m\) is a possibly singular weight function of indefinite sign. The authors show the existence of an unbounded sequence of positive variational eigenvalues via a standard min-max formula of Lusternik-Schnirelman type, and characterize the first and second eigenvalues by means of a Rayleigh quotient and symmetric orbits on a weighted \(L^p\)-sphere. The monotonicity properties of the first and second eigenvalues with respect to the weight function \(m\) are also investigated.
Reviewer: Qin Dongdong (Changsha)Existence of steady solutions for a general model for micropolar electrorheological fluid flowshttps://zbmath.org/1517.351782023-09-22T14:21:46.120933Z"Kaltenbach, Alex"https://zbmath.org/authors/?q=ai:kaltenbach.alex"Růžička, Michael"https://zbmath.org/authors/?q=ai:ruzicka.michaelSummary: In this paper, we study the existence of solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted Sobolev spaces. By deploying the Lipschitz truncation technique, we establish the existence of solutions without additional assumptions on the electric field.A Liouville type result and quantization effects on the system \(-\Delta u = u J'(1-|u|^2)\) for a potential convex near zerohttps://zbmath.org/1517.352192023-09-22T14:21:46.120933Z"De Maio, Umberto"https://zbmath.org/authors/?q=ai:de-maio.umberto"Hadiji, Rejeb"https://zbmath.org/authors/?q=ai:hadiji.rejeb"Lefter, Catalin"https://zbmath.org/authors/?q=ai:lefter.catalin-george"Perugia, Carmen"https://zbmath.org/authors/?q=ai:perugia.carmenSummary: We consider a Ginzburg-Landau type equation in \(\mathbb{R}^2\) of the form \(-\Delta u = u J'(1-|u|^2)\) with a potential function \(J\) satisfying weak conditions allowing for example a zero of infinite order in the origin. We extend in this context the results concerning quantization of finite potential solutions of \textit{H. Brézis} et al. [Arch. Ration. Mech. Anal. 126, No. 1, 35--58 (1994; Zbl 0809.35019)] who treat the case when \(J\) behaves polinomially near \(0,\) as well as a result of Th. Cazenave, found in the same reference, and concerning the form of finite energy solutions.The heterogeneous Helmholtz problem with spherical symmetry: Green's operator and stability estimateshttps://zbmath.org/1517.352232023-09-22T14:21:46.120933Z"Sauter, Stefan"https://zbmath.org/authors/?q=ai:sauter.stefan-a"Torres, Céline"https://zbmath.org/authors/?q=ai:torres.celineThis article examines the modeling of wave propagation phenomena in the frequency domain using the Helmholtz equation in heterogeneous media, with a specific focus on media with discontinuous and highly oscillating wave speed. The authors specifically analyze problems with spherical symmetry and provide explicit representations of the Green's operator, as well as stability estimates in terms of both frequency and wave speed.
Reviewer: Hongyu Liu (Hong Kong)Fractional Hardy equations with critical and supercritical exponentshttps://zbmath.org/1517.352362023-09-22T14:21:46.120933Z"Bhakta, Mousomi"https://zbmath.org/authors/?q=ai:bhakta.mousomi"Ganguly, Debdip"https://zbmath.org/authors/?q=ai:ganguly.debdip"Montoro, Luigi"https://zbmath.org/authors/?q=ai:montoro.luigiSummary: We study the existence, nonexistence and qualitative properties of the solutions to the problem
\[
\begin{cases}
(-\Delta )^s u -\theta \frac{u}{|x|^{2s}} & =u^p - u^q \quad \text{in }\mathbb{R}^N \\
u & > 0 \quad \text{in } \mathbb{R}^N \\
u &\in \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N),
\end{cases}
\]
where \(s\in (0,1)\), \(N>2s\), \(q>p\ge{(N+2s)}/{(N-2s)}\), \(\theta \in (0, \Lambda_{N,s})\) and \(\Lambda_{N,s}\) is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions, we mean both the radial symmetry, that is obtained by using the moving plane method in a nonlocal setting on the whole \(\mathbb{R}^N\), and a suitable upper bound behavior of the solutions. To this last end, we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space.Existence and concentration result for fractional Choquard equations in \(\mathbb{R}^N\)https://zbmath.org/1517.352372023-09-22T14:21:46.120933Z"Che, Guofeng"https://zbmath.org/authors/?q=ai:che.guofeng"Su, Yu"https://zbmath.org/authors/?q=ai:su.yu"Chen, Haibo"https://zbmath.org/authors/?q=ai:chen.haiboSummary: In this paper, we consider the following fractional Choquard equations with competing potentials:
\[
\begin{cases}
\varepsilon^{2\alpha}(-\triangle)^\alpha u+V(x)u=\varepsilon^{\mu-N}f(x,u), \quad\text{ in }\mathbb{R}^N \\
u\in H^\alpha(\mathbb{R}^N),
\end{cases}
\]
where \(f(x,u)=\left(\int_{\mathbb{R}^N}\frac{K(y)|u(y)|^p}{|x-y|^\mu}\text{d}y\right) K(x)|u|^{p-2}u+\left(\int_{\mathbb{R}^N}\frac{Q(y)|u(y)|^q}{|x-y|^\mu}\text{d}y\right) Q(x)|u|^{q-2}u\), \(\varepsilon>0\) is a parameter, \(\alpha\in(0,1)\), \(0<\mu<N\), \(N>2\alpha\), \(\frac{2N-\mu }{N}<q<p<\frac{2N-\mu }{N-2\alpha}\), \(V(x)\) and \(K(x)\) are positive smooth functions with \(V(x)\) bounded from below, \(K(x)\) bounded, and \(Q(x)\) is a bounded function that allowed to be sign-changing. By using the minimax theorems and the Nehari manifold techniques, we establish the existence of a ground state solution \(u_\varepsilon\) for the above system when \(\varepsilon\) is small enough. Moreover, \(u_\varepsilon\) concentrates around a global minimum point of the least energy function as \(\varepsilon\to 0\).A fractional Ambrosetti-Prodi type problem in \(\mathbb{R}^N\)https://zbmath.org/1517.352392023-09-22T14:21:46.120933Z"de Lima, Romildo N."https://zbmath.org/authors/?q=ai:de-lima.romildo-n"Torres Ledesma, César E."https://zbmath.org/authors/?q=ai:torres-ledesma.cesar-e"Nóbrega, Alânnio B."https://zbmath.org/authors/?q=ai:nobrega.alannio-bSummary: In this paper we deal with the existence and non-existence of solutions for the following Ambrosetti-Prodi type problem
\[
\begin{cases}
(-\Delta)^s u=P(x)\Big( g(u)+f(x)\Big) \text{ in } \mathbb{R}^N, \\
u \in D^s (\mathbb{R}^N), \lim_{|x|\rightarrow +\infty}u(x)=0,
\end{cases}
\tag{P}
\]
where \(N > 2s, s \in (0,1), P\in C(\mathbb{R}^N,\mathbb{R}^+), f\in C^{1,\sigma}(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N), g\in C^{1,\sigma} (\mathbb{R})\) and \((-\Delta)^s u\) is the fractional Laplacian. The main tools used are the sub-supersolution method and Leray-Schauder topological degree theory.Quantitative weighted bounds for Littlewood-Paley functions generated by fractional heat semigroups related with Schrödinger operatorshttps://zbmath.org/1517.352492023-09-22T14:21:46.120933Z"Yang, Li"https://zbmath.org/authors/?q=ai:yang.li.6"Li, Pengtao"https://zbmath.org/authors/?q=ai:li.pengtao(no abstract)Critical fractional \(p\)-Laplacian system with negative exponentshttps://zbmath.org/1517.352512023-09-22T14:21:46.120933Z"Zhu, Qinghao"https://zbmath.org/authors/?q=ai:zhu.qinghao"Qi, Jianming"https://zbmath.org/authors/?q=ai:qi.jianming(no abstract)Normalized solutions to the fractional Schrödinger equation with potentialhttps://zbmath.org/1517.352522023-09-22T14:21:46.120933Z"Zuo, Jiabin"https://zbmath.org/authors/?q=ai:zuo.jiabin"Liu, Chungen"https://zbmath.org/authors/?q=ai:liu.chungen"Vetro, Calogero"https://zbmath.org/authors/?q=ai:vetro.calogeroSummary: This paper is concerned with the existence of normalized solutions to a class of Schrödinger equations driven by a fractional operator with a parametric potential term. We obtain minimization of energy functional associated with that equations assuming basic conditions for the potential. Our work offers a partial extension of previous results to the non-local case.Monotonicity-based shape reconstruction for an inverse scattering problem in a waveguidehttps://zbmath.org/1517.352542023-09-22T14:21:46.120933Z"Arens, Tilo"https://zbmath.org/authors/?q=ai:arens.tilo"Griesmaier, Roland"https://zbmath.org/authors/?q=ai:griesmaier.roland"Zhang, Ruming"https://zbmath.org/authors/?q=ai:zhang.rumingSummary: We consider an inverse medium scattering problem for the Helmholtz equation in a closed cylindrical waveguide with penetrable compactly supported scattering objects. We develop novel monotonicity relations for the eigenvalues of an associated modified near field operator, and we use them to establish linearized monotonicity tests that characterize the support of the scatterers in terms of near field observations of the corresponding scattered waves. The proofs of these shape characterizations rely on the existence of localized wave functions, which are solutions to the scattering problem in the waveguide that have arbitrarily large norm in some prescribed region, while at the same time having arbitrarily small norm in some other prescribed region. As a byproduct we obtain a uniqueness result for the inverse medium scattering problem in the waveguide with a simple proof. Some numerical examples are presented to document the potentials and limitations of this approach.A Borg-Levinson theorem for magnetic Schrödinger operators on a Riemannian manifoldhttps://zbmath.org/1517.352552023-09-22T14:21:46.120933Z"Bellassoued, Mourad"https://zbmath.org/authors/?q=ai:bellassoued.mourad"Choulli, Mourad"https://zbmath.org/authors/?q=ai:choulli.mourad"Dos Santos Ferreira, David"https://zbmath.org/authors/?q=ai:dos-santos-ferreira.david"Kian, Yavar"https://zbmath.org/authors/?q=ai:kian.yavar"Stefanov, Plamen"https://zbmath.org/authors/?q=ai:stefanov.plamen-dLet \(M\) be a compact, smooth Riemannian manifold with boundary. Sufficient conditions are provided under which boundary spectral data of a Laplacian on \(M\) with magnetic and electric potential determines these potentials uniquely. Concretely, if the eigenvalues of two such operators, with Dirichlet boundary conditions, as well as the normal derivatives on the boundary for the corresponding normalized eigenfunctions are sufficiently close to each other in a certain sense, then their magnetic potentials coincide, and under some slightly stronger assumptions the same is true for the electric potentials. Also Neumann boundary conditions are considered. Moreover, stability of this inverse problem is studied. A general underlying assumption is that \(M\) is simple, meaning that its boundary is strictly convex and at each point \(x \in M\) the exponential map \(\exp_x : \exp_x^{-1} (M) \to M\) is a diffeomorphism.
Reviewer: Jonathan Rohleder (Stockholm)Active control of scalar Helmholtz fields in the presence of known impenetrable obstacleshttps://zbmath.org/1517.352562023-09-22T14:21:46.120933Z"Besabe, Lander"https://zbmath.org/authors/?q=ai:besabe.lander"Onofrei, Daniel"https://zbmath.org/authors/?q=ai:onofrei.danielSummary: In this paper, we consider the question of actively manipulating scalar Helmholtz fields radiated by a given source that is supported on a compact domain. We claim that the field radiated by the source approximates given scalar fields in prescribed exterior regions while maintaining desired far field patterns in prescribed directions in the presence of exterior known impenetrable obstacles. For simplicity of the exposition, we consider a simplified geometry with only one obstacle, one region of control, and a finite number of far field directions and present a theoretical argument for our claim stated above. Afterwards, we also show how it can be elementarily extended to the general case. Further, we construct a numerical scheme to compute these boundary inputs using the method of moments, the addition theorem, Tikhonov regularization, and Laplace spherical functions.
For the entire collection see [Zbl 1512.35009].Stability estimate for an inverse problem of a hyperbolic heat equation from boundary measurementhttps://zbmath.org/1517.352612023-09-22T14:21:46.120933Z"Jbalia, A."https://zbmath.org/authors/?q=ai:jbalia.aymenSummary: We are concerned with an inverse problem arising in thermal imaging in a bounded domain \(\Omega\subset\mathbb{R}^n\), \(n=2,3\). This inverse problem consists in the determination of the heat exchange coefficient \(q(x)\) appearing in the boundary of a hyperbolic heat equation with Robin boundary condition. A double-logarithmic stability estimate is developed.Inverse problems for anisotropic obstacle problems with multivalued convection and unbalanced growthhttps://zbmath.org/1517.352702023-09-22T14:21:46.120933Z"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengda"Bai, Yunru"https://zbmath.org/authors/?q=ai:bai.yunru"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-dSummary: The prime goal of this paper is to introduce and study a highly nonlinear inverse problem of identification discontinuous parameters (in the domain) and boundary data in a nonlinear variable exponent elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is formulated the sum of a weighted anisotropic \(p\)-Laplacian and a weighted anisotropic \(q\)-Laplacian (called the weighted anisotropic \((p,q)\)-Laplacian), a multivalued reaction term depending on the gradient, two multivalued boundary conditions and an obstacle constraint. We, first, employ the theory of nonsmooth analysis and a surjectivity theorem for pseudomonotone operators to prove the existence of a nontrivial solution of the anisotropic elliptic obstacle problem, which relies on the first eigenvalue of the Steklov eigenvalue problem for the \(p_-\)-Laplacian. Then, we introduce the parameter-to-solution map for the anisotropic elliptic obstacle problem, and establish a critical convergence result of the Kuratowski type to parameter-to-solution map. Finally, a general framework is proposed to examine the solvability of the nonlinear inverse problem.Half-space solutions with \(7/2\) frequency in the thin obstacle problemhttps://zbmath.org/1517.352722023-09-22T14:21:46.120933Z"Savin, Ovidiu"https://zbmath.org/authors/?q=ai:savin.ovidiu-v"Yu, Hui"https://zbmath.org/authors/?q=ai:yu.huiThis paper concerns classification of \(7/2\)-homogeneous solutions for the thin obstacle problem in \(\mathbb{R}^3\). More precisely, let \(\mathcal{P}\) denote the family of \(7/2\)-homogeneous solutions and let \(\mathcal{F}\subseteq \mathcal{P}\) be the set of half-space homogeneous solutions to the thin obstacle problem in \(\mathbb{R}^3\). The authors show that \(\mathcal{F}\) forms an isolated family in \(\mathcal{P}\). Furthermore, the authors quantify the rate of convergence of the blow-ups if one of the blow-up limits is in \(\mathcal{F}\). The results are proven through an interesting improvement of flatness type argument around equilibriums which do not correspond to the lowest frequency.
Reviewer: Wenhui Shi (Clayton)Existence of solutions for inclusion problems in Musielak-Orlicz-Sobolev space settinghttps://zbmath.org/1517.352732023-09-22T14:21:46.120933Z"Dong, Ge"https://zbmath.org/authors/?q=ai:dong.ge"Fang, Xiaochun"https://zbmath.org/authors/?q=ai:fang.xiaochun(no abstract)Geometric harmonic analysis I. A sharp divergence theorem with nontangential pointwise traceshttps://zbmath.org/1517.420012023-09-22T14:21:46.120933Z"Mitrea, Dorina"https://zbmath.org/authors/?q=ai:mitrea.dorina"Mitrea, Irina"https://zbmath.org/authors/?q=ai:mitrea.irina"Mitrea, Marius"https://zbmath.org/authors/?q=ai:mitrea.mariusThe present book is the first in a series of five volumes, at the confluence of Harmonic Analysis,
Geometric Measure Theory, Function Space Theory, and Partial Differential Equations. The series is generically
branded as Geometric Harmonic Analysis, with the individual volumes carrying the following subtitles:
Volume I: A Sharp Divergence Theorem with Nontangential Pointwise Traces;
Volume II: Function Spaces Measuring Size and Smoothness on Rough Sets;
Volume III: Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering;
Volume IV: Boundary Layer Potentials in Uniformly Rectifiable Domains, and Applications to Complex Analysis;
Volume V: Fredholm Theory and Finer Estimates for Integral Operators, with Applications to Boundary Problems.
The present review is concerned with the first volume. In the first chapter, starting from the classical work
of De Giorgi-Federer, the authors develop a new generation of divergence theorems both in the Euclidean space
as well as in the setting of Riemannian manifolds. The most striking feature is that the vector field in question is
strictly defined in the underlying open set and its boundary trace is considered in a pointwise nontangential fashion.
In the second chapter, a wealth of examples and counter-examples are presented, indicating that the main results are
optimal from a variety of perspectives.
The third chapter gathers foundational material from measure theory and topology.
Chapter four contains a variety of selected topics from (or inspired by) distribution theory.
For example, the authors develop a brand of distribution theory on arbitrary subsets of the Euclidean space, taking
Lipschitz functions with bounded support as test functions. Here they also coin the notion of
``bullet product'' which, in essence, is a weak version (modeled upon integration by parts) of the inner product of
the normal vector to a domain with a given vector field satisfying only some very mild integrability properties in that domain.
Among other things, a proof of Leibniz's product rule for weak derivatives is provided, and the chapter ends with what the
authors call the contribution at infinity of a vector field.
In the fifth chapter, the author discusses basic results from Geometric Measure Theory, including thick sets,
the corkscrew condition, the geometric measure theoretic boundary, area and coarea formulas, countable rectifiability,
approximate tangent planes, functions of bounded variation, sets of locally finite perimeter, Ahlfors regularity,
uniformly rectifiable sets, the local John condition, and nontangentially accessible domains.
The sixth chapter is focused on tools from Harmonic Analysis, such as the regularized distance function,
Whitney's Extension Theorem, and the fractional Hardy-Littlewood maximal operator in non-metric settings.
This chapter also contains an informative review of Clifford algebras (which are higher-dimensional versions
of the field of complex numbers, that happen to be highly non-commutative, in which a brand of complex analysis may be developed),
and a discussion of reverse Hölder inequalities and interior estimates. The authors close this chapter by introducing
the solid maximal function and defining maximal Lebesgue spaces.
The seventh chapter, entitled Quasi-Metric Spaces and Spaces of Homogeneous Type, consists of the following sections:
Quasi-Metric Spaces and a Sharp Metrization Result; Estimating Integrals Involving the Quasi-Distance;
Hölder Spaces on Quasi-Metric Spaces; Functions of Bounded Mean Oscillations on Spaces of Homogeneous Type;
Whitney Decompositions on Geometrically Doubling Quasi-Metric Spaces; The Hardy-Littlewood Maximal Operator
on Spaces of Homogeneous Type; Muckenhoupt Weights on Spaces of Homogeneous Type; The Fractional Integration Theorem.
The eighth chapter is entitled Open Sets with Locally Finite Surface Measures and Boundary Behavior.
The first section focuses on nontangential approach regions in arbitrary open sets. The second and third sections
deal with the basic properties of the nontangential maximal operator. The fourth section contains size estimates
for the nontangential maximal operator involving a doubling measure, while the fifth one is reserved for a
comparison between the nontangential and tangential maximal operators. In the sixth section, the authors establish
off-diagonal Carleson measure estimates of reverse Hölder type, which are crucial ingredients in the proofs of the main results.
The seventh section elaborates on estimates for Marcinkiewicz type integrals and applications. The eighth and the ninth sections
are on what the authors call the nontangentially accessible boundary and, respectively, the nontangential boundary trace operator.
The tenth section treats the averaged nontangential maximal operator.
The last chapter contains the proofs of the main results pertaining to the family of divergence theorems stated in chapter one.
Reviewer: Mohammed El Aïdi (Bogotá)Spaces of Besov-Sobolev type and a problem on nonlinear approximationhttps://zbmath.org/1517.460242023-09-22T14:21:46.120933Z"Domínguez, Óscar"https://zbmath.org/authors/?q=ai:dominguez.oscar"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreas"Street, Brian"https://zbmath.org/authors/?q=ai:street.brian"Van Schaftingen, Jean"https://zbmath.org/authors/?q=ai:van-schaftingen.jean"Yung, Po-Lam"https://zbmath.org/authors/?q=ai:yung.polamLet \(\{\phi_j \}^\infty_{- \infty}\) be the usual dyadic resolution of unity in \(\mathbb R^n\), underlying the study of the well-known homogeneous spaces \(\dot{B}^s_{p,q}\) and \(\dot{F}^s_{p,q}\) in \(\mathbb R^n\). In particular,
\[
\|f \, | \dot{B}^s_{p,q} \| = \Big( \sum^\infty_{j=-\infty} 2^{jsq} \big\| (\phi_j \hat{f} )^\vee \,| L_p (\mathbb R^n) \|^q \Big)^{1/q}
\]
is the standard quasi-norm in \(\dot{B}^s_{p,q}\). There are numerous modifications of these spaces. In the paper under review, the authors introduce spaces \(\mathcal{B}^s_p (\gamma, r)\), replacing \(L_p\) in \(B^s_{p,p}\) by \(L_{p,r} (\mu_\gamma)\), where \(L_{p,r}\) are Lorentz spaces and \(\mu_\gamma\) are special measures. In particular, \(\mathcal{B}^s_p (\gamma, p) = \dot{B}^s_{p,p}\), \(\gamma \in \mathbb R\). It is the main aim of this paper to study these spaces in detail, mainly parallel to related classical questions.
Reviewer: Hans Triebel (Jena)Elliptic operators with unbounded diffusion, drift and potential termshttps://zbmath.org/1517.470832023-09-22T14:21:46.120933Z"Boutiah, S. E."https://zbmath.org/authors/?q=ai:boutiah.sallah-eddine"Gregorio, F."https://zbmath.org/authors/?q=ai:gregorio.federica"Rhandi, A."https://zbmath.org/authors/?q=ai:rhandi.abdelaziz"Tacelli, C."https://zbmath.org/authors/?q=ai:tacelli.cristianSummary: We prove that the realization \(A_p\) in \(L^p(\mathbb{R}^N)\), \(1<p< \infty\), of the elliptic operator \(A=(1+|x|^\alpha)\Delta +b|x|^{\alpha-1}\frac{x}{|x|}\cdot \nabla -c|x|^{\beta}\) with domain \(D(A_p)=\{u \in W^{2,p}(\mathbb{R}^N)|Au \in L^p(\mathbb{R}^N)\}\) generates a strongly continuous analytic semigroup \(T(\cdot)\) provided that \(\alpha >2\), \(\beta > \alpha -2\) and any constants \(b \in \mathbb{R}\) and \(c>0\). This generalizes the recent results in [\textit{A. Canale} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 16, No. 2, 581--601 (2016; Zbl 1350.47033)] and in [\textit{G. Metafune} et al., Adv. Differ. Equ. 19, No. 5--6, 473--526 (2014; Zbl 1305.47029)]. Moreover, we show that \(T(\cdot)\) is consistent, immediately compact and ultracontractive.Perspectives in scalar curvature. Vol. 2https://zbmath.org/1517.530042023-09-22T14:21:46.120933ZPublisher's description: Volume I contains a long article by Misha Gromov based on his many years of involvement in this subject. It came from lectures delivered in Spring 2019 at IHES. There is some background given. Many topics in the field are presented, and many open problems are discussed. One intriguing point here is the crucial role played by two seemingly unrelated analytic means: index theory of Dirac operators and geometric measure theory.
Very recently there have been some real breakthroughs in the field. Volume I has several survey articles written by people who were responsible for these results.
For Volume II, many people in areas of mathematics and physics, whose work is somehow related to scalar curvature, were asked to write about this in any way thay pleased. This gives rise to a wonderful collection of articles, some with very broad and historical views, others which discussed specific fascinating subjects.
These two books give a rich and powerful view of one of geometry's very appealing sides.
The articles of this volume will be reviewed individually. For Vol. 1 see [Zbl 1516.53002].
Indexed articles:
\textit{Stolz, Stephan}, Positive scalar curvature -- constructions and obstructions, 5-49 [Zbl 07733267]
\textit{Botvinnik, Boris; Rosenberg, Jonathan}, Positive scalar curvature on \(\mathrm{Pin}^\pm\)- and \(\mathrm{Spin}^c\)-manifolds and manifolds with singularities, 51-81 [Zbl 07733268]
\textit{Botvinnik, Boris; Ebert, Johannes}, Positive scalar curvature and homotopy theory, 83-157 [Zbl 07733269]
\textit{Min-Oo, Maung}, The Lichnerowicz formula and lower bounds for the scalar curvature, 161-199 [Zbl 07733270]
\textit{Zhang, Weiping}, Deformed Dirac operators and scalar curvature, 201-214 [Zbl 07733271]
\textit{Chodosh, Otis; Li, Chao}, Recent results concerning topological obstructions to positive scalar curvature, 215-230 [Zbl 07733272]
\textit{Dranishnikov, Alexander}, Positive scalar curvature, macroscopic dimension, and inessential manifolds, 231-248 [Zbl 07733273]
\textit{Herzlich, Marc}, Scalar curvature, mass, and other asymptotic invariants, 249-311 [Zbl 07733274]
\textit{Wang, Jian}, Topological characterization of contractible 3-manifolds with positive scalar curvature, 313-321 [Zbl 07733275]
\textit{Bär, Christian; Hanke, Bernhard}, Boundary conditions for scalar curvature, 325-377 [Zbl 07733276]
\textit{Richard, Thomas; Zhu, Jintian}, Small two spheres in positive scalar curvature, using minimal hypersurfaces, 381-415 [Zbl 07733277]
\textit{Calegari, Danny; Marques, Fernando C.; Neves, André}, Minimal surface entropy of negatively curved manifolds, 417-428 [Zbl 07733278]
\textit{Galloway, Gregory J.}, Marginally outer trapped surfaces and scalar curvature rigidity, 429-449 [Zbl 07733279]
\textit{Hijazi, Oussama; Montiel, Sebastián; Raulot, Simon}, Scalar curvature, spinors, eigenvalues, and mass, 453-487 [Zbl 07733280]
\textit{De Lima, Levi Lopes}, Conserved quantities in general relativity: the case of initial datasets with a non-compact boundary, 489-518 [Zbl 07733281]
\textit{Ammann, Bernd; Glöckle, Jonathan}, Dominant energy condition and spinors on Lorentzian manifolds, 519-592 [Zbl 07733282]
\textit{Bray, Hubert; Hirsch, Sven; Kazaras, Demetre; Khuri, Marcus; Zhang, Yiyue}, Spacetime harmonic functions and applications to mass, 593-639 [Zbl 07733283]
\textit{Sormani, Christina}, Conjectures on convergence and scalar curvature, 645-722 [Zbl 07733284]
\textit{Hu, Xue; Shi, Yuguang}, Geometric aspects of quasi-local mass and Gromov's fill-in problem, 723-738 [Zbl 07733285]
\textit{Miao, Pengzi}, Interpreting mass via Riemannian polyhedra, 739-759 [Zbl 07733286]
\textit{Guo, Hao; Xie, Zhizhang; Yu, Guoliang}, Quantitative K-theory, positive scalar curvature, and bandwidth, 763-798 [Zbl 07733287]
\textit{Liokumovich, Yevgeny; Maximo, Davi}, Waist inequality for 3-manifolds with positive scalar curvature, 799-831 [Zbl 07733288]
\textit{Benameur, Moulay-Tahar}, The Gromov-Lawson index and the Baum-Connes assembly map, 835-882 [Zbl 07733289]Sharp pointwise Weyl laws for Schrödinger operators with singular potentials on flat torihttps://zbmath.org/1517.580062023-09-22T14:21:46.120933Z"Huang, Xiaoqi"https://zbmath.org/authors/?q=ai:huang.xiaoqi"Zhang, Cheng"https://zbmath.org/authors/?q=ai:zhang.chengSummary: The Weyl law of the Laplacian on the flat torus \({\mathbb{T}}^n\) is concerning the number of eigenvalues \(\le \lambda^2\), which is equivalent to counting the lattice points inside the ball of radius \(\lambda\) in \({\mathbb{R}}^n\). The leading term in the Weyl law is \(c_n\lambda^n\), while the sharp error term \(O(\lambda^{n-2})\) is only known in dimension \(n\ge 5\). Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. This result establishes the sharpness of the general theorems for the Schrödinger operators \(H_V=-\Delta_g+V\) in the previous work [Adv. Math. 410 A, Article ID 108688, 34 p. (2022; Zbl 1501.58015)] of the authors, and extends the 3-dimensional results of \textit{R. L. Frank} and \textit{J. Sabin} [Pure Appl. Anal. 5, No. 1, 85--144 (2023; Zbl 1514.35300)] to any dimensions by using a different approach. Our approach is a combination of Fourier analysis techniques on the flat torus, Li-Yau's heat kernel estimates, Blair-Sire-Sogge's eigenfunction estimates, and Duhamel's principle for the wave equation.Dirichlet boundary value problems for elliptic operators with measure datahttps://zbmath.org/1517.600972023-09-22T14:21:46.120933Z"Yang, Saisai"https://zbmath.org/authors/?q=ai:yang.saisai"Zhang, Tusheng"https://zbmath.org/authors/?q=ai:zhang.tusheng-sSummary: In this paper, we prove that there exists a unique weak solution to the Dirichlet boundary value problem for second order elliptic operators whose coefficients are signed measures. Our approach is probabilistic, which also gives a representation of the solution. The heat kernel estimates and the theory of additive functionals play a crucial role in our approach.Wavenumber explicit convergence of a multiscale generalized finite element method for heterogeneous Helmholtz problemshttps://zbmath.org/1517.650852023-09-22T14:21:46.120933Z"Chupeng, Ma"https://zbmath.org/authors/?q=ai:chupeng.ma"Alber, Christian"https://zbmath.org/authors/?q=ai:alber.christian"Scheichl, Robert"https://zbmath.org/authors/?q=ai:scheichl.robertSummary: In this paper, a generalized finite element (FE) method with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors of carefully designed local eigenvalue problems defined on generalized harmonic spaces. At both continuous and discrete levels, (i) wavenumber explicit and nearly exponential decay rates for local and global approximation errors are obtained without any assumption on the size of subdomains and (ii) a quasi-optimal convergence of the method is established by assuming that the size of subdomains is \(O(1/k)\) (\(k\) is the wavenumber). A novel resonance effect between the wavenumber and the dimension of local spaces on the decay of error with respect to the oversampling size is implied by the analysis. Furthermore, for fixed dimensions of local spaces, the discrete local errors are proved to converge as \(h\rightarrow 0\) (\(h\) denoting the mesh size) toward the continuous local errors. The method at the continuous level extends the plane wave partition of unity method [\textit{I. Babuška} and \textit{J. M. Melenk}, Int. J. Numer. Methods Eng. 40, No. 4, 727--758 (1997; Zbl 0949.65117)] to the heterogeneous-coefficients case, and at the discrete level, it delivers an efficient noniterative domain decomposition method for solving discrete Helmholtz problems resulting from standard FE discretizations. Numerical results are provided to confirm the theoretical analysis and to validate the proposed method.Generalized weak Galerkin finite element methods for biharmonic equationshttps://zbmath.org/1517.651162023-09-22T14:21:46.120933Z"Li, Dan"https://zbmath.org/authors/?q=ai:li.dan.2"Wang, Chunmei"https://zbmath.org/authors/?q=ai:wang.chunmei"Wang, Junping"https://zbmath.org/authors/?q=ai:wang.junpingSummary: The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete \(H^2\) norm and a \(L^2\) norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation.Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichmenthttps://zbmath.org/1517.651172023-09-22T14:21:46.120933Z"Smetana, Kathrin"https://zbmath.org/authors/?q=ai:smetana.kathrin"Taddei, Tommaso"https://zbmath.org/authors/?q=ai:taddei.tommasoSummary: We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations. CB-pMOR is designed to deal with large-scale problems for which full-order solves are not affordable in a reasonable time frame or parameters' variations induce topology changes that prevent the application of monolithic pMOR techniques. We rely on the partition-of-unity method to devise global approximation spaces from local reduced spaces, and on Galerkin projection to compute the global state estimate. We propose a randomized data compression algorithm based on oversampling for the construction of the components' reduced spaces: the approach exploits random boundary conditions of controlled smoothness on the oversampling boundary. We further propose an adaptive residual-based enrichment algorithm that exploits global reduced-order solves on representative systems to update the local reduced spaces. We prove exponential convergence of the enrichment procedure for linear coercive problems; we further present numerical results for a two-dimensional nonlinear diffusion problem to illustrate the many features of our methodology and demonstrate its effectiveness.Scattering resonances in unbounded transmission problems with sign-changing coefficienthttps://zbmath.org/1517.780022023-09-22T14:21:46.120933Z"Carvalho, Camille"https://zbmath.org/authors/?q=ai:carvalho.camille"Moitier, Zoïs"https://zbmath.org/authors/?q=ai:moitier.zoisSummary: It is well known that classical optical cavities can exhibit localized phenomena associated with scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the `quasimodes to resonances' argument from the black box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, e.g. made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and shows that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial's properties, scattering resonances situated close to the real axis are associated with surface plasmons. Examples for several metamaterial cavities are provided.Out-of-plane enhancement in a discrete random halfspacehttps://zbmath.org/1517.780042023-09-22T14:21:46.120933Z"McCargar, Reid K."https://zbmath.org/authors/?q=ai:mccargar.reid-k"Lang, Roger H."https://zbmath.org/authors/?q=ai:lang.roger-henryThe authors develop the heuristic radiative transfer theory concerning waves propagating in a random medium modeled by particles independently and identically distributed throughout the slab. The model is based on the Helmholtz equation. The authors investigate the mean-wave Green's function.
Reviewer: Vladimir Mityushev (Kraków)Nearest-neighbor approximation in one-excitation state evolution along spin-1/2 chain governed by \(XX\)-Hamiltonianhttps://zbmath.org/1517.810682023-09-22T14:21:46.120933Z"Fel'dman, E. B."https://zbmath.org/authors/?q=ai:feldman.eduard-b"Zenchuk, A. I."https://zbmath.org/authors/?q=ai:zenchuk.alexandre-iSummary: The approximation of nearest neighbor interaction (NNI) is widely used in short-time spin dynamics with dipole-dipole interactions (DDI) when the intensity of spin-spin interaction is \(\sim 1/r^3\), where \(r\) is a distance between those spins. However, NNI can not approximate the long time evolution in such systems. We consider the system with the intensity of the spin-spin interaction \(\sim 1/r^\alpha\), \(\alpha\geq3\), and find the low boundary \(\alpha_c\) of applicability of the NNI to the evolution of an arbitrary one-excitation initial quantum state in the homogeneous spin chain governed by the \(XX\)-Hamiltonian. We obtain the logarithmic dependence of \(\alpha_c\) on the chain length.