Recent zbMATH articles in MSC 31Bhttps://zbmath.org/atom/cc/31B2024-02-28T19:32:02.718555ZWerkzeugA unified trapezoidal quadrature method for singular and hypersingular boundary integral operators on curved surfaceshttps://zbmath.org/1527.310022024-02-28T19:32:02.718555Z"Wu, Bowei"https://zbmath.org/authors/?q=ai:wu.bowei"Martinsson, Per-Gunnar"https://zbmath.org/authors/?q=ai:martinsson.per-gunnarSummary: This paper describes a locally corrected trapezoidal quadrature method for the discretization of singular and hypersingular boundary integral operators (BIOs) that arise in solving boundary value problems for elliptic partial differential equations. The quadrature is based on a uniform grid in parameter space coupled with the standard punctured trapezoidal rule. A key observation is that the error incurred by the singularity in the kernel can be expressed exactly using generalized Euler-Maclaurin formulas that involve the Riemann zeta function in 2 dimensions (2D) and the Epstein zeta functions in 3 dimensions (3D). These expansions are exploited to correct the errors via local stencils at the singular point using a novel systematic moment-fitting approach. This new method provides a unified treatment of all common BIOs (Laplace, Helmholtz, Stokes, etc.). We present numerical examples that show convergence of up to 32nd-order in 2D and 9th-order in 3D with respect to the mesh size.Khavinson problem for hyperbolic harmonic mappings in Hardy spacehttps://zbmath.org/1527.310062024-02-28T19:32:02.718555Z"Chen, Jiaolong"https://zbmath.org/authors/?q=ai:chen.jiaolong"Kalaj, David"https://zbmath.org/authors/?q=ai:kalaj.david"Melentijević, Petar"https://zbmath.org/authors/?q=ai:melentijevic.petarSummary: In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that \(u=\mathcal{P}_{\Omega}[\phi]\) and \(\phi \in L^p(\partial{\Omega},\mathbb{R})\), where \(p\in [1,\infty]\), \(\mathcal{P}_{\Omega}[\phi]\) denotes the Poisson integral of \(\phi\) with respect to the hyperbolic Laplacian operator \(\Delta_h\) in \(\Omega \), and \(\Omega\) denotes the unit ball \(\mathbb{B}^n\) or the half-space \(\mathbb{H}^n \). For any \(x \in \Omega\) and \(l\in \mathbb{S}^{n-1} \), let \(\mathbf{C}_{\Omega,q}(x)\) and \(\mathbf{C}_{\Omega,q}(x;l)\) denote the optimal numbers for the gradient estimate
\[
\left |\nabla u(x)\right |\leq \mathbf{C}_{\Omega ,q}(x)\left \|\phi \right \|_{L^p(\partial{\Omega}, \mathbb{R})}
\]
and the gradient estimate in the direction \(l\)
\[
\left |\langle \nabla u(x),l\rangle \right |\leq \mathbf{C}_{\Omega,q}(x;l)\left \|\phi \right \|_{L^p(\partial{\Omega},\mathbb{R})},
\]
respectively. Here \(q\) is the conjugate of \(p\). If \(q\in [1,\infty]\), then \(\mathbf{C}_{\mathbb{B}^n,q}(0)\equiv \mathbf{C}_{\mathbb{B}^n,q}(0;l)\) for any \(l\in \mathbb{S}^{n-1} \). If \(q=\infty \), \(q = 1\) or \(q\in [\frac{2K_0-1}{n-1}+1,\frac{2K_0}{n-1}+1]\) with \(K_0\in \mathbb{N} \), then \(\mathbf{C}_{\mathbb{B}^n,q}(x)=\mathbf{C}_{\mathbb{B}^n,q}(x;\pm \frac{x}{|x|})\) for any \(x\in \mathbb{B}^n\setminus \{0\} \), and \(\mathbf{C}_{\mathbb{H}^n,q}(x)=\mathbf{C}_{\mathbb{H}^n,q}(x;\pm e_n)\) for any \(x\in \mathbb{H}^n \). However, if \(q\in (1,\frac{n}{n-1})\), then \(\mathbf{C}_{\mathbb{B}^n,q}(x)=\mathbf{C}_{\mathbb{B}^n,q}(x;t_x)\) for any \(x\in \mathbb{B}^n\setminus \{0\} \), and \(\mathbf{C}_{\mathbb{H}^n,q}(x)=\mathbf{C}_{\mathbb{H}^n,q}(x;t_{e_n})\) for any \(x\in \mathbb{H}^n \). Here \(t_w\) denotes any unit vector in \(\mathbb{R}^n\) such that \(\langle t_w,w \rangle = 0\) for \(w\in \mathbb{R}^n\setminus \{0\} \).Stability of elliptic Harnack inequalitieshttps://zbmath.org/1527.310072024-02-28T19:32:02.718555Z"Chen, Zhen-Qing"https://zbmath.org/authors/?q=ai:chen.zhen-qingSummary: We survey some recent progress in the study of stability of elliptic Harnack inequalities under form-bounded perturbations for strongly local Dirichlet forms on complete locally compact separable metric spaces.Upper bounds for the number of isolated critical points via the Thom-Milnor theoremhttps://zbmath.org/1527.310082024-02-28T19:32:02.718555Z"Zolotov, Vladimir"https://zbmath.org/authors/?q=ai:zolotov.vladimirSummary: We apply the Thom-Milnor theorem to obtain the upper bounds on the amount of isolated (1) critical points of a potential generated by several fixed point charges (Maxwell's problem on point charges), (2) critical points of SINR, (3) critical points of a potential generated by several fixed Newtonian point masses augmented with a quadratic term, (4) central configurations in the \(n\)-body problem. In particular, we get an exponential bound for Maxwell's problem and the polynomial bound for the case of an ``even dimensional'' potential in Maxwell's problem.Classes of kernels and continuity properties of the double layer potential in Hölder spaceshttps://zbmath.org/1527.310092024-02-28T19:32:02.718555Z"Lanza de Cristoforis, Massimo"https://zbmath.org/authors/?q=ai:lanza-de-cristoforis.massimoSummary: We prove the validity of regularizing properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a \textit{nonhomogeneous} second order elliptic differential operator with constant coefficients in Hölder spaces by exploiting an estimate on the maximal function of the tangential gradient with respect to the first variable of the kernel of the double layer potential and by exploiting specific imbedding and multiplication properties in certain classes of kernels of integral operators and a generalization of a result for integral operators on differentiable manifolds.Fourier method for the Neumann problem on a torushttps://zbmath.org/1527.310102024-02-28T19:32:02.718555Z"Ashtab, Z."https://zbmath.org/authors/?q=ai:ashtab.z"Morais, J."https://zbmath.org/authors/?q=ai:morais.joao-pedro"Porter, R. Michael"https://zbmath.org/authors/?q=ai:porter.r-michaelSummary: The Fourier method approach to the Neumann problem for the Laplacian operator in the case of a solid torus contrasts in many respects with the much more straight forward situation of a ball in 3-space. Although the Dirichlet-to-Neumann map can be readily expressed in terms of series expansions with toroidal harmonics, we show that the resulting equations contain undetermined parameters which cannot be calculated algebraically. A method for rapidly computing numerical solutions of the Neumann problem is presented with numerical illustrations. The results for interior and exterior domains combine to provide a solution for the Neumann problem for the case of a shell between two tori.Quasianalytic functionals and ultradistributions as boundary values of harmonic functionshttps://zbmath.org/1527.310112024-02-28T19:32:02.718555Z"Debrouwere, Andreas"https://zbmath.org/authors/?q=ai:debrouwere.andreas"Vindas, Jasson"https://zbmath.org/authors/?q=ai:vindas.jassonSummary: We study boundary values of harmonic functions in spaces of quasianalytic functionals and spaces of ultradistributions of non-quasianalytic type. As an application, we provide a new approach to Hörmander's support theorem for quasianalytic functionals. Our main technical tool is a description of ultradifferentiable functions by almost harmonic functions, a concept that we introduce in this article. We work in the setting of ultradifferentiable classes defined via weight matrices. In particular, our results simultaneously apply to the two standard classes defined via weight sequences and via weight functions.Stability for an inverse spectral problem of the biharmonic Schrödinger operatorhttps://zbmath.org/1527.310122024-02-28T19:32:02.718555Z"Yao, Xiaohua"https://zbmath.org/authors/?q=ai:yao.xiaohua"Zhao, Yue"https://zbmath.org/authors/?q=ai:zhao.yueSummary: Let \(V\) be a bounded potential. We prove a Hölder stability estimate of determining the potential \(V\) from the boundary spectral data of the biharmonic operator \(\Delta^2 + V\). The boundary spectral data consists of the Dirichlet eigenvalues \(\lambda_k\) and the normal derivatives of the eigenfunctions \(\partial_\nu \phi_k, \partial_\nu \Delta \phi_k\) on the boundary of a bounded domain. The analysis depends on the asymptotic behavior of the spectral data.Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable setshttps://zbmath.org/1527.351492024-02-28T19:32:02.718555Z"David, G."https://zbmath.org/authors/?q=ai:david.guy"Mayboroda, S."https://zbmath.org/authors/?q=ai:mayboroda.svitlanaAuthors' abstract: It was recently shown that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a domain with an \(n -1\) dimensional uniformly rectifiable boundary, in the presence of now well-understood additional topological constraints. The topological restrictions, while mild, are necessary, as the counterexamples of \textit{C. J. Bishop} and \textit{P. W. Jones} [Ann. Math. (2) 132, No. 3, 511--547 (1990; Zbl 0726.30019)] show, and no analogues of these results have been available for higher co-dimensional sets. In the present paper, we show that for any \(d<n-1\) and for any domain with a \(d\)-dimensional uniformly rectifiable boundary the elliptic measure of an appropriate degenerate elliptic operator is absolutely continuous with respect to the Hausdorff measure of the boundary. There are no topological or dimensional restrictions contrary to the aforementioned results.
Reviewer: Lavi Karp (Karmiel)Asymptotic behavior of generalized capacities with applications to eigenvalue perturbations: the higher dimensional casehttps://zbmath.org/1527.351912024-02-28T19:32:02.718555Z"Abatangelo, Laura"https://zbmath.org/authors/?q=ai:abatangelo.laura"Léna, Corentin"https://zbmath.org/authors/?q=ai:lena.corentin"Musolino, Paolo"https://zbmath.org/authors/?q=ai:musolino.paoloSummary: We provide a full series expansion of a generalization of the so-called \(u\)-capacity related to the Dirichlet-Laplacian in dimension three and higher, extending the results of
\textit{L. Abatangelo} et al. [ESAIM, Control Optim. Calc. Var. 27, Paper No. S25, 43 p. (2021; Zbl 1468.35100); J. Funct. Anal. 283, No. 12, Article ID 109718, 50 p. (2022; Zbl 1507.35126)]
dealing with the planar case. We apply the result in order to study the asymptotic behavior of perturbed eigenvalues when Dirichlet conditions are imposed on a small regular subset of the domain of the eigenvalue problem.A note on the equivalence and the boundary behavior of a class of Sobolev capacitieshttps://zbmath.org/1527.490072024-02-28T19:32:02.718555Z"Christof, Constantin"https://zbmath.org/authors/?q=ai:christof.constantin"Müller, Georg"https://zbmath.org/authors/?q=ai:muller.georg.1|muller.georgSummary: The purpose of this paper is to study different notions of Sobolev capacity commonly used in the analysis of obstacle- and Signorini-type variational inequalities. We review basic facts from capacity theory in an abstract setting that is tailored to the study of \(W^{1,p}\)- and \(W^{1-1}/^{p,p}\)-capacities, and we prove equivalency results that relate several approaches found in the literature to each other. Motivated by applications in contact mechanics, we especially focus on the behavior of different Sobolev capacities on and near the boundary of the domain in question. As a result, we obtain, for example, that the most common approaches to the sensitivity analysis of Signorini-type problems are exactly the same.Quantum oscillator as a minimization problemhttps://zbmath.org/1527.810472024-02-28T19:32:02.718555Z"D'Eliseo, Maurizio M."https://zbmath.org/authors/?q=ai:deliseo.maurizio-m(no abstract)