Recent zbMATH articles in MSC 58Jhttps://zbmath.org/atom/cc/58J2023-01-20T17:58:23.823708ZWerkzeugSymmetric ground state solutions for the Choquard Logarithmic equation with exponential growthhttps://zbmath.org/1500.340132023-01-20T17:58:23.823708Z"Yuan, Shuai"https://zbmath.org/authors/?q=ai:yuan.shuai"Chen, Sitong"https://zbmath.org/authors/?q=ai:chen.sitongSummary: We investigate the existence of ground state solutions for the fractional Choquard Logarithmic equation
\[
(- \Delta)^{1 / 2} u + V (x) u + (\ln | \cdot | \ast | u |^2) u = f (u), \quad x \in \mathbb{R},
\]
where \(V \in \mathcal{C} (\mathbb{R}, [ 0, \infty))\) and the \(f\) satisfies exponential critical growth. The present paper extends and complements the result of \textit{E. S. Böer} and \textit{O. H. Miyagaki} [``The Choquard logarithmic equation involving fractional Laplacian operator and a nonlinearity with exponential critical growth'', Preprint, \url{arXiv:2011.12806}]. In particular, our paper has two typical features. Firstly, using a weaker assumption on \(f\), we establish the energy inequality to exclude the vanishing case of the required Cerami sequence. Secondly, with the property of radial symmetry we shall use some new variational and analytic technique to establish our final result which is different to the arguments explored in [loc. cit.].Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcationshttps://zbmath.org/1500.350262023-01-20T17:58:23.823708Z"Duan, Daifeng"https://zbmath.org/authors/?q=ai:duan.daifeng"Niu, Ben"https://zbmath.org/authors/?q=ai:niu.ben"Wei, Junjie"https://zbmath.org/authors/?q=ai:wei.junjieSummary: We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.On a fractional Nirenberg problem via a non-degeneracy conditionhttps://zbmath.org/1500.351502023-01-20T17:58:23.823708Z"Alghanemi, Azeb"https://zbmath.org/authors/?q=ai:alghanemi.azeb"Rigane, Afef"https://zbmath.org/authors/?q=ai:rigane.afefSummary: In this paper, we establish existence results to a fractional Nirenberg problem of order \(\sigma\in(0,\frac{3}{2})\) on the standard three dimensional
sphere. Our approach of this paper consists of studying the limits of the non-compact orbits of the gradient flow, as did Bahri for the Nirenberg problem, and performing a Morse Lemma at infinity near these ends. This involves computing their topological contributions which then yields the results.Nonnegative solution of a class of double phase problems with logarithmic nonlinearityhttps://zbmath.org/1500.351792023-01-20T17:58:23.823708Z"Aberqi, Ahmed"https://zbmath.org/authors/?q=ai:aberqi.ahmed"Benslimane, Omar"https://zbmath.org/authors/?q=ai:benslimane.omar"Elmassoudi, Mhamed"https://zbmath.org/authors/?q=ai:elmassoudi.mhamed"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraSummary: This manuscript proves the existence of a nonnegative, nontrivial solution to a class of double-phase problems involving potential functions and logarithmic nonlinearity in the setting of Sobolev space on complete manifolds. Some applications are also being investigated. The arguments are based on the Nehari manifold and some variational techniques.On a class of systems of hyperbolic equations describing pseudo-spherical or spherical surfaceshttps://zbmath.org/1500.352062023-01-20T17:58:23.823708Z"Kelmer, Filipe"https://zbmath.org/authors/?q=ai:kelmer.filipe"Tenenblat, Keti"https://zbmath.org/authors/?q=ai:tenenblat.ketiSummary: We consider systems of partial differential equations of the form
\[
\begin{cases}
u_{x t} = F (u, u_x, v, v_x),\\
v_{x t} = G (u, u_x, v, v_x),
\end{cases}
\]
describing pseudospherical (\textbf{pss}) or spherical surfaces (\textbf{ss}), meaning that, their generic solutions \((u(x, t), v(x, t))\) provide metrics, with coordinates \((x, t)\), on open subsets of the plane, with constant curvature \(K = - 1\) or \(K = 1\). These systems can be described as the integrability conditions of \(\mathfrak{g}\)-valued linear problems, with \(\mathfrak{g} = \mathfrak{sl}(2, \mathbb{R})\) or \(\mathfrak{g} = \mathfrak{su}(2)\), when \(K = - 1\), \(K = 1\), respectively. We obtain characterization and also classification results. Applications of the theory provide new examples and new families of systems of differential equations, which also contain generalizations of a Pohlmeyer-Lund-Regge type system and of the Konno-Oono coupled dispersionless system. We provide explicitly the first few conservation laws, from an infinite sequence, for some of the systems describing \textbf{pss}.Inverse scattering for critical semilinear wave equationshttps://zbmath.org/1500.352202023-01-20T17:58:23.823708Z"Sá Barreto, Antônio"https://zbmath.org/authors/?q=ai:sa-barreto.antonio"Uhlmann, Gunther"https://zbmath.org/authors/?q=ai:uhlmann.gunther-a"Wang, Yiran"https://zbmath.org/authors/?q=ai:wang.yiranSummary: We show that the scattering operator for defocusing energy critical semilinear wave equations \(\square u+f(u)=0\), \(f\in C^\infty(\mathbb{R})\) and \(f\sim u^5\), in three space dimensions, determines \(f\).Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groupshttps://zbmath.org/1500.370302023-01-20T17:58:23.823708Z"Yamaguchi, Yoshikazu"https://zbmath.org/authors/?q=ai:yamaguchi.yoshikazu.1|yamaguchi.yoshikazuAuthor's abstract: We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral expression of the Ruelle zeta function. This integral expression is derived from the functional equation of the Selberg zeta function for a discrete subgroup with elliptic elements in \(\mathrm{PSL}_2(\mathbb{R})\). We also show that the asymptotic behavior of the higher-dimensional Reidemeister torsion is determined by the contribution of the identity element to the integral expression of the Ruelle zeta function.
Reviewer: Jialun Li (Zürich)Towards semi-classical analysis for sub-elliptic operatorshttps://zbmath.org/1500.430062023-01-20T17:58:23.823708Z"Fischer, Véronique"https://zbmath.org/authors/?q=ai:fischer.veroniqueSummary: We discuss the recent developments of semi-classical and micro-local analysis in the context of nilpotent Lie groups and for sub-elliptic operators. In particular, we give an overview of pseudo-differential calculi recently defined on nilpotent Lie groups as well as of the notion of quantum limits in the Euclidean and nilpotent cases.
For the entire collection see [Zbl 1487.35004].Transfer operators, induced probability spaces, and random walk modelshttps://zbmath.org/1500.471232023-01-20T17:58:23.823708Z"Jorgensen, P."https://zbmath.org/authors/?q=ai:jorgensen.palle-e-t"Tian, F."https://zbmath.org/authors/?q=ai:tian.fengSummary: We study a family of discrete-time random-walk models. The starting point is a fixed generalized transfer operator \(R\) subject to a set of axioms, and a given endomorphism in a compact Hausdorff space \(X\). Our setup includes a host of models from applied dynamical systems, and it leads to general path-space probability realizations of the initial transfer operator. The analytic data in our construction is a pair \((h,\lambda)\), where \(h\) is an \(R\)-harmonic function on \(X\), and \(\lambda\) is a given positive measure on \(X\) subject to a certain invariance condition defined from \(R\). With this we show that there are then discrete-time random-walk realizations in explicit path-space models; each associated to a probability measures \(\mathbb P\) on path-space, in such a way that the initial data allows for spectral characterization: The initial endomorphism in \(X\) lifts to an automorphism in path-space with the probability measure \(\mathbb P\) quasi-invariant with respect to a shift automorphism. The latter takes the form of explicit multiresolutions in \(L^2\) of \(\mathbb P\) in the sense of Lax-Phillips scattering theory.Critical points of a mean field type functional on a closed Riemann surfacehttps://zbmath.org/1500.490042023-01-20T17:58:23.823708Z"Zhang, Mengjie"https://zbmath.org/authors/?q=ai:zhang.mengjie"Yang, Yunyan"https://zbmath.org/authors/?q=ai:yang.yunyanSummary: Let \((\Sigma,g)\) be a closed Riemann surface and \(H^1(\Sigma)\) be the usual Sobolev space. For any real number \(\rho \), we define a generalized mean field type functional \(J_{\rho,\phi}\colon H^1(\Sigma)\rightarrow \mathbb{R}\) by
\[
J_{\rho,\phi}(u)=\frac{1}{2} \bigg(\int_{\Sigma}|\nabla_g u|^2 d v_g+\int_{\Sigma}\phi (u-\overline{u}) d v_g \bigg)-\rho\ln \int_{ \Sigma} h e^{u-\overline{u}} d v_g,
\]
where \(h\colon \Sigma\to\mathbb{R}\) is a smooth positive function, \(\phi\colon \mathbb{R}\to\mathbb{R}\) is a smooth one-variable function and \(\overline{u}=\int_\Sigma u \, d v_g/|\Sigma|\). If \(\rho\in (8k\pi,8(k+1)\pi) (k\in \mathbb{N}^\ast), \phi\) satisfies \(|\phi(t)|\leq C (|t|^p+1) (1< p< 2)\) and \(|\phi^\prime(t)|\leq C (|t|^{p-1}+1)\) for some constant \(C\), then we get critical points of \(J_{\rho,\phi}\) by adapting min-max schemes of \textit{W. Ding} et al. [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16, No. 5, 653--666 (1999; Zbl 0937.35055)], \textit{Z. Djadli} [Commun. Contemp. Math. 10, No. 2, 205--220 (2008; Zbl 1151.53035)] and \textit{A. Malchiodi} [Discrete Contin. Dyn. Syst. 21, No. 1, 277--294 (2008; Zbl 1144.35372)].Li-Yau inequalities in geometric analysishttps://zbmath.org/1500.530012023-01-20T17:58:23.823708Z"Chow, Bennett"https://zbmath.org/authors/?q=ai:chow.bennettSummary: In this short survey article, we mention some of Peter Li's tremendous influence on geometric analysis, focusing on the development of Li-Yau inequalities and related ideas in geometric analysis and geometric flows. This started with the seminal 1986 Li-Yau paper, Hamilton's development for geometric flows, and in particular his miraculous matrix Harnack estimate for the Ricci flow, Perelman's differential Harnack estimate for the conjugate heat kernel under a Ricci flow background, and his related monotonicity formulas for Ricci flow, and Bamler's recent sharp gradient estimates, new monotonicity formulas, and their applications for Ricci flow.Notes on product semisymmetric connection in a locally decomposable Riemannian spacehttps://zbmath.org/1500.530292023-01-20T17:58:23.823708Z"Maksimovic, Miroslav"https://zbmath.org/authors/?q=ai:maksimovic.miroslav-d"Stankovic, Mica"https://zbmath.org/authors/?q=ai:stankovic.mica-sSummary: The purpose of this paper is to investigate the product semisymmetric connection in a locally decomposable Riemannian space. The curvature tensors of this connection were considered. Some properties of almost product structure, some properties of torsion tensor of product semisymmetric connection and some relations between curvature tensors and almost product structure are given. Also, the paper checks a special case of such connection when its generator is a gradient vector.Remarks on entropy formulae for linear heat equationhttps://zbmath.org/1500.530972023-01-20T17:58:23.823708Z"Ji, Yucheng"https://zbmath.org/authors/?q=ai:ji.yuchengSummary: In this note, we prove some new entropy formula for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type equations.The relative index theorem for general first-order elliptic operatorshttps://zbmath.org/1500.580082023-01-20T17:58:23.823708Z"Bandara, Lashi"https://zbmath.org/authors/?q=ai:bandara.lashiSummary: The relative index theorem is proved for general first-order elliptic operators that are complete and coercive at infinity over measured manifolds. This extends the original result by Gromov-Lawson for generalised Dirac operators as well as the result of Bär-Ballmann for Dirac-type operators. The theorem is seen through the point of view of boundary value problems, using the graphical decomposition of elliptically regular boundary conditions for general first-order elliptic operators due to Bär-Bandara. Splitting, decomposition and the Phi-relative index theorem are proved on route to the relative index theorem.Higher localised \(\hat{A}\)-genera for proper actions and applicationshttps://zbmath.org/1500.580092023-01-20T17:58:23.823708Z"Guo, Hao"https://zbmath.org/authors/?q=ai:guo.hao"Mathai, Varghese"https://zbmath.org/authors/?q=ai:mathai.vargheseSummary: For a finitely generated discrete group \(\Gamma\) acting properly on a spin manifold \(M\), we formulate new topological obstructions to \(\Gamma \)-invariant metrics of positive scalar curvature on \(M\) that take into account the cohomology of the classifying space \(\underline{B}{\Gamma}\) for proper actions.
In the cocompact case, this leads to a natural generalisation of Gromov-Lawson's notion of higher \(\hat{A} \)-genera to the setting of proper actions by groups with torsion. It is conjectured that these invariants obstruct the existence of \(\Gamma \)-invariant positive scalar curvature on \(M\). For classes arising from the subring of \(H^\ast(\underline{B}{\Gamma}, \mathbb{R})\) generated by elements of degree at most 2, we are able to prove this, under suitable assumptions, using index-theoretic methods for projectively invariant Dirac operators and a twisted \(L^2\)-Lefschetz fixed-point theorem involving a weighted trace on conjugacy classes. The latter generalises a result of \textit{B.-L. Wang} and \textit{H. Wang} [J. Differ. Geom. 102, No. 2, 285--349 (2016; Zbl 1348.58014)] to the projective setting. In the special case of free actions and the trivial conjugacy class, this reduces to a theorem of \textit{V. Mathai} [Contemp. Math. 231, 203--225 (1999; Zbl 0942.46042)], which provided a partial answer to a conjecture of Gromov-Lawson on higher \(\hat{A}\)-genera. If \(M\) is non-cocompact, we obtain obstructions to \(M\) being a partitioning hypersurface inside a non-cocompact \(\Gamma \)-manifold with non-negative scalar curvature that is positive in a neighbourhood of the hypersurface. Finally, we define a quantitative version of the twisted higher index, as first introduced in [\textit{H. Guo} et al., ``Quantitative \(K\)-theory, positive scalar curvature, and band width'', Preprint, \url{arXiv:2010.01749}], and use it to prove a parameterised vanishing theorem in terms of the lower bound of the total curvature term in the square of the twisted Dirac operator.Mapping properties for operator-valued pseudodifferential operators on toroidal Besov spaceshttps://zbmath.org/1500.580102023-01-20T17:58:23.823708Z"Barraza Martínez, B."https://zbmath.org/authors/?q=ai:martinez.b-barraza|barraza-martinez.bienvenido"Denk, R."https://zbmath.org/authors/?q=ai:denk.robert"Hernández Monzón, J."https://zbmath.org/authors/?q=ai:monzon.j-hernandez|hernandez-monzon.jairo"Nendel, M."https://zbmath.org/authors/?q=ai:nendel.maxToroidal pseudodifferential calculus on the \(n\)-dimensional torus \(\mathbb{T}^n\) introduced in \textit{M. Ruzhansky} and \textit{V. Turunen}'s monograph [Pseudo-differential operators and symmetries. Background analysis and advanced topics. Basel: Birkhäuser (2010; Zbl 1193.35261)] provides us with a global quantization procedure on \(\mathbb{T}^n\). Mapping properties for such toroidal pseudodifferential operators on various kinds of function spaces such as \(L^p\), Besov and Hölder spaces have been investigated by several scholars in the scalar-valued case.
In this article, the authors study toroidal pseudodifferential operators associated with symbols with values in the space of bounded linear operators on some Banach space \(E\). Furthermore, they prove mapping properties for such pseudodifferential operators on toroidal Besov spaces with coefficients in \(E\). The symbol class used throughout the article is the class of Hörmander symbols of limited smoothness with respect to the space variable. In addition, the dyadic decomposition of the covariable space \(\mathbb{Z}^n\) is the key ingredient of the construction of \(E\)-valued Besov spaces utilized in the article. The proof of mapping properties for toroidal pseudodifferential operators between these Besov spaces is based on the convolution kernel representation of a given pseudodifferential operator using a dyadic decomposition.
Reviewer: Gihyun Lee (Ghent)Uniform resolvent estimates on manifolds of bounded curvaturehttps://zbmath.org/1500.580112023-01-20T17:58:23.823708Z"Smith, Hart F."https://zbmath.org/authors/?q=ai:smith.hart-fAuthor's abstract: We establish \(L^{q*}\rightarrow L^q\) bounds for the resolvent of the Laplacian on compact Riemannian manifolds assuming only that the sectional curvatures of the manifold are uniformly bounded. When the resolvent parameter lies outside a parabolic neighborhood of \([0,\infty)\), the operator norm of the resolvent is shown to depend only on upper bounds for the sectional curvature and diameter and lower bounds for the volume. The resolvent bounds are derived from square-function estimates for the wave equation, an approach that admits the use of paradifferential approximations in the parametrix construction.
Reviewer: Themistocles M. Rassias (Athína)Fractal uncertainty principle and its applications (after Bourgain, Dyatlov, Jin, Nonnenmacher, Zahl)https://zbmath.org/1500.810512023-01-20T17:58:23.823708Z"Dang, Nguyen Viet"https://zbmath.org/authors/?q=ai:dang.nguyen-vietSummary: In this talk, we will describe a new uncertainty principle that forbids to an \(L^2\) function to be localized simultaneously in space and in frequency near fractal sets satisfying certain porosity assumptions. In a second place, we will discuss spectacular applications of this principle to problems in geometric analysis on hyperbolic surfaces.
For the entire collection see [Zbl 1486.00041].\textit{pp}-wave initial datahttps://zbmath.org/1500.830092023-01-20T17:58:23.823708Z"García-Parrado, Alfonso"https://zbmath.org/authors/?q=ai:garcia-parrado-gomez-lobo.alfonsoSummary: An \textit{initial data characterization} for vacuum \textit{pp}-wave spacetimes in dimension four is constructed. This is a vacuum initial data set plus some extra conditions guaranteeing that the data development is a subset of a vacuum \textit{pp}-wave. Some of the extra conditions only depend on the same quantities used to construct the vacuum initial data, namely the \textit{first} and the \textit{second fundamental forms} while others are related to a \textit{conformal Killing initial data characterization} (CKID).Transversely trapping surfaces: dynamical versionhttps://zbmath.org/1500.830112023-01-20T17:58:23.823708Z"Yoshino, Hirotaka"https://zbmath.org/authors/?q=ai:yoshino.hirotaka"Izumi, Keisuke"https://zbmath.org/authors/?q=ai:izumi.keisuke"Shiromizu, Tetsuya"https://zbmath.org/authors/?q=ai:shiromizu.tetsuya"Tomikawa, Yoshimune"https://zbmath.org/authors/?q=ai:tomikawa.yoshimuneSummary: We propose new concepts, a dynamically transversely trapping surface (DTTS) and a marginally DTTS, as indicators for a strong gravity region. A DTTS is defined as a two-dimensional closed surface on a spacelike hypersurface such that photons emitted from arbitrary points on it in transverse directions are acceleratedly contracted in time, and a marginally DTTS is reduced to the photon sphere in spherically symmetric cases. (Marginally) DTTSs have a close analogy with (marginally) trapped surfaces in many aspects. After preparing the method of solving for a marginally DTTS in the time-symmetric initial data and the momentarily stationary axisymmetric initial data, some examples of marginally DTTSs are numerically constructed for systems of two black holes in the Brill-Lindquist initial data and in the Majumdar-Papapetrou spacetimes. Furthermore, the area of a DTTS is proved to satisfy the Penrose-like inequality \(A_0\le 4\pi (3GM)^2\), under some assumptions. Differences and connections between a DTTS and the other two concepts proposed by us previously, a loosely trapped surface [\textit{T. Shiromizu} et al., PTEP, Prog. Theor. Exper. Phys. 2017, No. 3, Article ID 033E01, 6 p. (2017; Zbl 1477.83009)] and a static/stationary transversely trapping surface [\textit{H. Yoshino} et al., PTEP, Prog. Theor. Exper. Phys. 2017, No. 6, Article ID 063E01, 23 p. (2017; Zbl 1477.83070)], are also discussed. A (marginally) DTTS provides us with a theoretical tool to significantly advance our understanding of strong gravity fields. Also, since DTTSs are located outside the event horizon, they could possibly be related with future observations of strong gravity regions in dynamical evolutions.A Schwarzian on the stretched horizonhttps://zbmath.org/1500.830132023-01-20T17:58:23.823708Z"Carlip, S."https://zbmath.org/authors/?q=ai:carlip.stevenSummary: It is well known that the Euclidean black hole action has a boundary term at the horizon proportional to the area. I show that if the horizon is replaced by a stretched horizon with appropriate boundary conditions, a new boundary term appears, described by a Schwarzian action similar to the recently discovered boundary actions in ``nearly anti-de Sitter'' gravity.A mechanism of baryogenesis for causal fermion systemshttps://zbmath.org/1500.830312023-01-20T17:58:23.823708Z"Finster, Felix"https://zbmath.org/authors/?q=ai:finster.felix"Jokel, Maximilian"https://zbmath.org/authors/?q=ai:jokel.maximilian"Paganini, Claudio F."https://zbmath.org/authors/?q=ai:paganini.claudio-fSummary: It is shown that the theory of causal fermion systems gives rise to a novel mechanism of baryogenesis. This mechanism is worked out computationally in globally hyperbolic spacetimes in a way which enables the quantitative study in concrete cosmological situations.Self-consistent equation for torsion arising as a consequence of the Dirac sea quantum fluctuations in external classical electromagnetic and gravitational fieldshttps://zbmath.org/1500.830442023-01-20T17:58:23.823708Z"Vergeles, S. N."https://zbmath.org/authors/?q=ai:vergeles.sergei-nikitovichSummary: The quantum fluctuations of the Dirac field in external classical gravitational and electromagnetic fields are studied. A self-consistent equation for torsion is calculated, which is obtained using one-loop fermion diagrams.