Recent zbMATH articles in MSC 31https://zbmath.org/atom/cc/312022-11-17T18:59:28.764376ZWerkzeugDiscrepancy of minimal Riesz energy pointshttps://zbmath.org/1496.111022022-11-17T18:59:28.764376Z"Marzo, Jordi"https://zbmath.org/authors/?q=ai:marzo.jordi"Mas, Albert"https://zbmath.org/authors/?q=ai:mas.albertThis paper is concerned with upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz \(s\)-energy and the logarithmic energy on the sphere \({\mathbb S}^d\).
Recall, that a spherical cap \(D_r(x)\) is the set \( \{y \in{\mathbb S}^d\ :\ |x-y|<r\}, \) where \(x\in {\mathbb S}^d \), \(r\in[0,2]\), and the spherical cap discrepancy of a point set \(X\subset {\mathbb S}^d\) is
\[
\sup_{r,x}\Big|\frac{\#\big(X\cap D_r(x)\big)}{\# X}-\tilde{\sigma}(D_r(x))\Big|,
\]
where \(\tilde{\sigma}\) is the uniform probability measure on \({\mathbb S}^d\). The Riesz/logarithmic energy of a set \(V\subset {\mathbb S}^d\) is given by
\[
E_s(V)=\sum_{\substack{x,y\in V \\
x\neq y}}R_s(x,y),\hspace{0.4cm}\mbox{ where }\hspace{0.4cm} R_s(x,y)=\begin{cases} |x-y|^{-s}, &\mbox{for }0<s<d\\
-\log(|x-y|),&\mbox{for }s=0.\end{cases}
\]
Let \(V_N\) denote all subsets of \({\mathbb S}^d\) with \(N\)-elements, where \(N\in {\mathbb N}\). Since the kernel \(R_s(x,y)\) is lower semi-continuous and \({\mathbb S}^d\) is compact, there exists a minimizing configuration \(\omega^s_N=\{x_1,\ldots,x_N\}\in V_N\) for the energy, for every \(N\in {\mathbb N}\). Let \(\chi_A\) denote the characteristic function of the set \(A\), i.e., \(\chi_A(x)=1\) if \(x\in A\) and \(\chi_A(x)=0\) else. The authors show the following upper bound:
Theorem 1.1. Let \(\omega^s_N=\{x_1,\ldots,x_N\}\) be the \(N\)-point minimizer of the Riesz or logarithmic energy, then
\[
c_{s,d}\cdot\sup_{r,x}\Big|\frac{\#\big(\omega^s_N\cap D_r(x)\big)}{N}-\tilde{\sigma}(D_r(x))\Big|\ \leq\ \chi_{[0,d-2]}(s)\cdot N^{-\frac{2}{d(d-s+1)}}+\chi_{(d-2,d)}(s)\cdot N^{-\frac{2(d-s)}{d(d-s+4)}},
\]
with a constant \(c_{s,d}\) that depends on \(d,s\) only.
This theorem is derived by relating the spherical cap discrepancy with a notion of discrepancy involving Sobolev norms (Theorem 1.5 and Proposition 5.2).
The behavior of following integral operator on \(L^2\big({\mathbb S}^d\big)\) is investigated (Section 2 and Section 3):
\[
R_s(f)=\int_{{\mathbb S}^d}R_s(x,y)f(y)d\tilde{\sigma}(y),
\]
where, among other things, it is shown (in Proposition 2.2) that \(R_s\) diagonalizes in the standard basis of spherical harmonics, and its eigenvalues are computed as well as their asymptotic behavior. Further, some formal identities were derived (in Remark 2.3) for \(R_s(x,y)\) in terms of Gegenbauer, also known as ultraspherical polynomials, which might be of independent interest.
The authors derive a differential equation to relate \(R_s(x,x_0)\) to \(R_{s+2}(x,x_0)\) for \(x\neq x_0\) (in Lemma 2.5) via the Laplace-Beltrami operator, which is later used to show superharmonicity of \(R_s(x,x_0)\) near \(x_0\) (in Lemma 3.1), and to obtain an upper bound (in Corollary 3.7) of
\[
\frac{1}{N^2}\sum_{j\neq k}\int_{D_j}\int_{D_k}R_s(x,y)d\tilde{\sigma}(x)d\tilde{\sigma}(y),
\]
where each \(D_j\) is a small disc around \(x_j\in\omega^s_N \). This result is then applied to derive the upper bound for the Sobolev discrepancy (in Theorem 1.5).
This paper generalizes an unpublished result due to \textit{T. Wolff} [``Fekete points on spheres'', Preprint]. The conclusion of Theorem 1.1 (partially) improves upon results, obtained by Kleiner, Sjögren, Götz and Brauchart.
Some typos were found on page 479, where the average of a function \(f\) over a disc \(D\) should be integrated over \(D\); on page 499 we find \(R_s(x-y)\) instead of \(R_s(x,y)\).
Reviewer: Damir Ferizović (Leuven)Moduli of doubly connected domains under univalent harmonic mapshttps://zbmath.org/1496.310012022-11-17T18:59:28.764376Z"Bshouty, Daoud"https://zbmath.org/authors/?q=ai:bshouty.daoud-h"Lyzzaik, Abdallah"https://zbmath.org/authors/?q=ai:lyzzaik.abdallah"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.antti"Vasudevarao, Allu"https://zbmath.org/authors/?q=ai:vasudevarao.alluLet \(\mathcal{T}(t)=\mathbb{C}\setminus([-1,1]\cup[t,\infty))\), \(t>1\), be a Teichmüller doubly connected domain. The Teichmüller-Nitsche problem is formulated as follows:
For which values \(s,t\), \(1<s,t<\infty\), does a harmonic homeomorphism \(f:\mathcal{T}(s)\rightarrow\mathcal{T}(t)\) exist?
In the paper, the Teichmüller-Nitsche problem is solved for symmetric harmonic homeomorphisms between \(\mathcal{T}(s)\) and \(\mathcal{T}(t)\). This problem is solved by using the method of extremal length.
The following question suggested by \textit{T. Iwaniec} et al. [Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 5, 1017--1030 (2011; Zbl 1267.30059)] is also considered:
Characterize pairs \((\Omega,\Omega^*)\) of doubly connected domains that admit a univalent harmonic mapping from \(\Omega\) onto \(\Omega^*\).
This question is tested regarding the moduli of the doubly connected domains related by harmonic homeomorphisms. The paper concludes with relevant questions.
Reviewer: Konstantin Malyutin (Kursk)Boundary points of angular type form a set of zero harmonic measurehttps://zbmath.org/1496.310022022-11-17T18:59:28.764376Z"Gardiner, Stephen J."https://zbmath.org/authors/?q=ai:gardiner.stephen-j"Sjödin, Tomas"https://zbmath.org/authors/?q=ai:sjodin.tomasThe authors show that, if \(\Omega\) is a subset of \(\mathbb{R}^N\), then the set \(AT(\Omega)\) of boundary points \(x_0 \in \partial \Omega\) of angular type has zero harmonic measure. This result provides an answer to a problem presented by Dvoretzky (see Problem 7.15 of [\textit{W. K. Hayman}, in: Symp. complex Analysis, Canterbury 1973, 143--180 (1974; Zbl 0354.30001)]).
Reviewer: Paolo Musolino (Padova)Neumann boundary condition for a nonlocal biharmonic equationhttps://zbmath.org/1496.310032022-11-17T18:59:28.764376Z"Turmetov, Batirkhan Khudaĭbergenovich"https://zbmath.org/authors/?q=ai:turmetov.batirkhan-khudaybergenovich"Karachik, Valeriĭ Valentinovich"https://zbmath.org/authors/?q=ai:karachik.valery-vSummary: The solvability conditions for a class of boundary value problems for a nonlocal biharmonic equation in the unit ball with the Neumann conditions on the boundary are studied. The nonlocality of the equation is generated by some orthogonal matrix. The presence and uniqueness of a solution to the proposed Neumann boundary condition is examined, and an integral representation of the solution to the Dirichlet problem in terms of the Green's function for the biharmonic equation in the unit ball is obtained.
First, some auxiliary statements are established: the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given, the representation of the solution to the Dirichlet problem in terms of this Green's function is written, and the values of the integrals of the functions perturbed by the orthogonal matrix are found. Then a theorem for the solution to the auxiliary Dirichlet problem for a nonlocal biharmonic equation in the unit ball is proved. The solution to this problem is written using the Green's function of the Dirichlet problem for the regular biharmonic equation. An example of solving a simple problem for a nonlocal biharmonic equation is given. Next, we formulate a theorem on necessary and sufficient conditions for the solvability of the Neumann boundary condition for a nonlocal biharmonic equation. The main theorem is proved based on two lemmas, with the help of which it is possible to transform the solvability conditions of the Neumann boundary condition to a simpler form. The solution to the Neumann boundary condition is presented through the solution to the auxiliary Dirichlet problem.Constructing discrete harmonic functions in wedgeshttps://zbmath.org/1496.310042022-11-17T18:59:28.764376Z"Hoang, Viet Hung"https://zbmath.org/authors/?q=ai:hoang.viet-hung"Raschel, Kilian"https://zbmath.org/authors/?q=ai:raschel.kilian"Tarrago, Pierre"https://zbmath.org/authors/?q=ai:tarrago.pierreThis article proposes a systematic method for the construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, the authors prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.
Reviewer: Marius Ghergu (Dublin)The mutual singularity of harmonic measure and Hausdorff measure of codimension smaller than onehttps://zbmath.org/1496.310052022-11-17T18:59:28.764376Z"Tolsa, Xavier"https://zbmath.org/authors/?q=ai:tolsa.xavierThis article discusses the mutual singularity of the harmonic and Hausdorff measure of codimension smaller than one. Precisely, let \(\Omega\subset {\mathbb R}^{n+1}\), \(n\geq 2\), be an open set, \(E\subset \partial \Omega\) and \(s\in (n, n+1)\). Suppose that:
(i) There exists \(r_E>0\) and \(c_E>0\) such that \(\mathrm{Cap}(B(x,r)\cap \Omega^c)\geq c_E r^{n-1}\) for all \(0<r\leq r_E\) and all \(x\in E\);
(ii) The harmonic measure and the \(s\)-Hausdorff measure on \(E\) are mutually absolute continuous.
The main result of the article establishes that under the above assumptions, both the harmonic measure and the \(s\)-Hausdorff measure of \(E\) are zero.
Reviewer: Marius Ghergu (Dublin)Martin boundary of killed random walks on isoradial graphshttps://zbmath.org/1496.310062022-11-17T18:59:28.764376Z"Boutillier, Cédric"https://zbmath.org/authors/?q=ai:boutillier.cedric"Raschel, Kilian"https://zbmath.org/authors/?q=ai:raschel.kilianThis article discusses killed planar random walks on isoradial graphs. Unlike the lattice case, isoradial graphs present some difficulties such as: not translation invariant, do not admit any group structure and are spatially non-homogeneous. In the current work, the authors compute the asymptotics of the Martin kernel, deduce the Martin boundary and show its minimality.
Reviewer: Marius Ghergu (Dublin)Boundary distance functions of Riemann domains over pre-Hilbert spaceshttps://zbmath.org/1496.320482022-11-17T18:59:28.764376Z"Abe, Makoto"https://zbmath.org/authors/?q=ai:abe.makoto"Honda, Tatsuhiro"https://zbmath.org/authors/?q=ai:honda.tatsuhiro"Shima, Tadashi"https://zbmath.org/authors/?q=ai:shima.tadashiSummary: In the present paper, we generalize subpluriharmonic functions in the sense of Fujita to infinite dimension. We show that, for every open set \(D\) in a complex normed space \(E\) and for the boundary distance function \(d\) of \(D\), the function \(-\ln d\) is subpluriharmonic on \(D\). Moreover, we show that, for every Riemann domain \((D, \pi)\) over a complex pre-Hilbert space \(E\) and for the boundary distance function \(d\) of \((D, \pi)\), the function \(-\ln d\) is locally subpluriharmonic on \(D\).The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flowshttps://zbmath.org/1496.320602022-11-17T18:59:28.764376Z"Guedj, Vincent"https://zbmath.org/authors/?q=ai:guedj.vincent"Lu, Chinh H."https://zbmath.org/authors/?q=ai:lu.chinh-h"Zeriahi, Ahmed"https://zbmath.org/authors/?q=ai:zeriahi.ahmedLet \(T>0\), \(\Omega\) be a bounded strictly pseudoconvex domain in \(\mathbb C^n\) and \(\Omega_T=(0,T)\times\Omega\). In this paper, the authors consider the following degenerate complex Monge-Ampère flow (CMAF), on \(\Omega_T\):
\[
dt\wedge(dd^cu)^n=e^{\partial_tu+F(t,z,u)}g(z)dt\wedge dV,
\]
where \(dV\) is the Euclidean volume on \(\mathbb C^n\), \(g\in L^p(\Omega)\) for some \(p>1\) and \(g>0\) a.e., \(F(t,z,r)\) is continuous on \([0,T)\times\Omega\times\mathbb R\) and bounded on \([0,T)\times\Omega\times J\) for each compact \(J\subset\mathbb R\), \(F\) is increasing in \(r\) and \((t,r)\to F(t,\cdot,r)\) is uniformly Lipschitz and semi-convex. This can be seen as a local version of the pluripotential Kähler-Ricci flow studied by the authors in the earlier work [Geom. Topol. 24, No. 3, 1225--1296 (2020; Zbl 1458.32035)].
To deal with the above equation (CMAF), the authors introduce and study the family \(\mathcal P(\Omega_T)\) of parabolic potentials. These are functions \(u:\Omega_T\to\mathbb R\cup\{-\infty\}\) whose slices \(u(t,\cdot)\) are plurisubharmonic and such that the family \(\{u(\cdot,z):\,z\in\Omega\}\) is locally uniformly Lipschitz in \((0,T)\). They prove in particular that the parabolic complex Monge-Ampère operator \(dt\wedge(dd^cu)^n\) is well defined as a measure for \(u\in\mathcal P(\Omega_T)\cap L^\infty_{\mathrm{loc}}(\Omega_T)\) and they introduce the notions of sub/super/solution to (CMAF) in this class. They also prove a comparison theorem for bounded parabolic potentials.
A function \(h\) defined on the parabolic boundary \(\big([0,T)\times\partial\Omega\big)\cup(\{0\}\times\Omega)\) of \(\Omega_T\) is a Cauchy-Dirichlet boundary data if \(h\) is continuous on \([0,T)\times\partial\Omega\), the family \(\{h(\cdot,z):\,z\in\partial\Omega\}\) is locally uniformly Lipschitz in \((0,T)\), \(h(0,\cdot)\in \operatorname{PSH}(\Omega)\cap L^\infty(\Omega)\) and \(\lim_{\Omega\ni z\to\zeta}h(0,z)=h(0,\zeta)\) for all \(\zeta\in\partial\Omega\).
The main result of the paper is that the Cauchy-Dirichlet problem for (CMAF) can be solved by the Perron method with parabolic potentials as admissible functions. Assuming that the Cauchy-Dirichlet boundary data \(h\) is such that for every \(S\in(0,T)\) there exists a constant \(C(S)\) with
\[
t|\partial_th(t,z)|\leq C(S) \text{ and } t^2\partial_t^2h(t,z)\leq C(S),\;\forall\,(t,z)\in (0,S]\times\partial\Omega,
\]
the authors prove that the pluripotential solution to this Cauchy-Dirichlet problem is given by the upper envelope of the family of all subsolutions with boundary data \(h\).
Reviewer: Dan Coman (Syracuse)On correct restrictions of bi-Laplace operator and their propertieshttps://zbmath.org/1496.351912022-11-17T18:59:28.764376Z"Koshanov, Bakhytbek D."https://zbmath.org/authors/?q=ai:koshanov.bakhytbek-d"Smatova, Gulzhazira D."https://zbmath.org/authors/?q=ai:smatova.gulzhazira-dSummary: The aim of the present paper is description of correct restrictions of bi-Laplace operators. As an example there are demonstrated correct problems for a non-homogeneous biharmonic equation with integral conditions.
For the entire collection see [Zbl 1436.46003].An inverse boundary problem for fourth-order Schrödinger equations with partial datahttps://zbmath.org/1496.351942022-11-17T18:59:28.764376Z"Duan, Zhi-Wen"https://zbmath.org/authors/?q=ai:duan.zhiwen"Han, Shuxia"https://zbmath.org/authors/?q=ai:han.shuxiaSummary: In this paper, we show that in dimension \(n\ge 3\), the knowledge of the Cauchy data for the fourth-order Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. The proof is based on the Carleman estimates and the construction of complex geometrical optics solutions.Failure of Fatou type theorems for solutions to PDE of \(p\)-Laplace type in domains with flat boundarieshttps://zbmath.org/1496.352142022-11-17T18:59:28.764376Z"Akman, Murat"https://zbmath.org/authors/?q=ai:akman.murat"Lewis, John"https://zbmath.org/authors/?q=ai:lewis.john-l"Vogel, Andrew"https://zbmath.org/authors/?q=ai:vogel.andrew-lIn [\textit{T. H. Wolff}, J. Anal. Math. 102, 371--394 (2007; Zbl 1213.35218)] (a posthumous publication of an unpublished result from 1984), highly oscillatory bounded solutions of \(div(\nabla u|\nabla u|^{p-2})=0\) in the semi-plane \(\mathbb R^2_+\) were constructed for \(p>2\) and Fatou's theorem was shown to fail for these solutions.
In this paper an analogue of the above result is obtained in domains of the form \(\mathbb R^n\setminus \Lambda_k\), where \(\Lambda_k\) is a \(k\)-dimensional subspace (\(1\leq k<n-1\)).
The same problem is also addressed for solutions to a more general class of PDE's modeled on the \(p\)-Laplacian, called \(\mathcal A\)-harmonic functions, but the obtained results are not as complete as in the \(p\)-harmonic case.
Reviewer: Eugenio Massa (São Carlos)Divergence \& curl with fractional orderhttps://zbmath.org/1496.354342022-11-17T18:59:28.764376Z"Liu, Liguang"https://zbmath.org/authors/?q=ai:liu.liguang"Xiao, Jie"https://zbmath.org/authors/?q=ai:xiao.jie.1Summary: This paper presents a novel analysis for Function Space Norms (F.S.N.) \& Partial Differential Equations (P.D.E.) within the fractional-nonlocal pair \(\{\operatorname{div}^*\mathbf{v},\operatorname{curl}^*\mathbf{v}\}\) that extends the classical-local pair \(\{\operatorname{div}\mathbf{v},\operatorname{curl}\mathbf{v}\}\) which has an inherent physical
content because of causing the conservation of mass \& the rotation produced by fluid elements in motion.The exact spectral asymptotic of the logarithmic potential on harmonic function spacehttps://zbmath.org/1496.460182022-11-17T18:59:28.764376Z"Vujadinović, Djordjije"https://zbmath.org/authors/?q=ai:vujadinovic.djordjijeSummary: In this paper we consider the product of the harmonic Bergman projection \(P_h:L^2(D)\rightarrow L^2_h(D)\) and the operator of logarithmic potential type defined by \(Lf(z)=-\frac{1}{2\pi}\int_D\ln|z-\xi|f(\xi)dA(\xi)\), where \(D\) is the unit disc in \(\mathbb{C}\). We describe the asymptotic behaviour of the eigenvalues of the operator \((P_hL)^\ast(P_hL)\). More precisely, we prove that
\[
\lim\limits_{n\to+\infty} n^2s_n(P_hL)=\sqrt{\frac{4\pi^2}{3}-1}.
\]Decay of harmonic functions for discrete time Feynman-Kac operators with confining potentialshttps://zbmath.org/1496.600872022-11-17T18:59:28.764376Z"Cygan, Wojciech"https://zbmath.org/authors/?q=ai:cygan.wojciech"Kaleta, Kamil"https://zbmath.org/authors/?q=ai:kaleta.kamil"Śliwiński, Mateusz"https://zbmath.org/authors/?q=ai:sliwinski.mateuszSummary: We propose and study a certain discrete time counterpart of the classical Feynman-Kac semigroup with a confining potential in a countably infinite space. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman-Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman-Kac operators. We include such examples as non-local discrete Schrödinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.Heat kernel estimates on spaces with varying dimensionhttps://zbmath.org/1496.600982022-11-17T18:59:28.764376Z"Ooi, Takumu"https://zbmath.org/authors/?q=ai:ooi.takumuSummary: We obtain sharp two-sided heat kernel estimates for some process whose regular Dirichlet form is strongly local on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume doubling property fails with respect to the measure induced by the associated Lebesgue measures. Thus the parabolic Harnack inequalities fail and the heat kernels do not enjoy Aronson type estimates. Our estimates show that the on-diagonal estimates are independent of the dimensions of the two parts of the space for small time, whereas they depend on their transience or recurrence for large time. These are multidimensional version of a space considered by \textit{Z.-Q. Chen} and \textit{S. Lou} [Ann. Probab. 47, No. 1, 213--269 (2019; Zbl 1466.60165)], in which a 1-dimensional space and a 2-dimensional space are connected at one point.New scheme of the discrete sources method for two-dimensional scattering problems by penetrable obstacleshttps://zbmath.org/1496.652302022-11-17T18:59:28.764376Z"Eremin, Yuri A."https://zbmath.org/authors/?q=ai:eremin.yuri-a"Tsitsas, Nikolaos L."https://zbmath.org/authors/?q=ai:tsitsas.nikolaos-l"Kouroublakis, Minas"https://zbmath.org/authors/?q=ai:kouroublakis.minas"Fikioris, George"https://zbmath.org/authors/?q=ai:fikioris.georgeSummary: A new computational scheme of the Discrete Sources Method (DSM) is developed for the numerical solution of two-dimensional acoustic and electromagnetic transmission boundary-value problems (BVPs) pertaining to the Helmholtz equation. To establish the new DSM scheme, we show that the matrix integral operator, corresponding to the transmission boundary conditions, has a dense range. The approximate solution of the BVP is constructed according to the new DSM scheme and it is proven that it converges uniformly to the exact solution of the BVP. An analytic representation is derived for the calculation of the scattering cross section in terms of the determined DSs amplitudes avoiding an integration over the unit circle. The numerical implementation procedure of the DSM is described and numerical results for the application of the DSM are presented. The new scheme is shown to be accurate and efficient even for highly-elongated scatterers.Traversable wormhole on the brane with non-exotic matter: a broader viewhttps://zbmath.org/1496.830442022-11-17T18:59:28.764376Z"Sengupta, Rikpratik"https://zbmath.org/authors/?q=ai:sengupta.rikpratik"Ghosh, Shounak"https://zbmath.org/authors/?q=ai:ghosh.shounak"Kalam, Mehedi"https://zbmath.org/authors/?q=ai:kalam.mehedi"Ray, Saibal"https://zbmath.org/authors/?q=ai:ray.saibal