Boundary behavior of solutions to elliptic equations in general domains
type: Item
instance of: scholarly article
title: Boundary behavior of solutions to elliptic equations in general domains
zbMATH Open document ID: 1409.35073
DOI: 10.4171/190
published in: EMS Tracts in Mathematics
publication date: 2018-10-10T00:00:00Z
review text: This book is about the Wiener criterion for elliptic partial differential equations of second-order and for polyharmonic equations. Also moduli of continuity are provided in irregular domains in the Euclidean \(n\)-dimensional space. The focus is not on smooth domains: the more irregular, the better. A boundary point is regular for the solution to the Laplace equation if and only if there exists a barrier function at that point. Perron's method provides a unique solution, which may fail to attain the correct boundary values at some points. Regularity of a boundary point \(\xi\in\partial\Omega\) means that, given any continuous boundary values \(g:\partial\Omega\to\mathbb{R}\), the solution to the Dirichlet problem \[ \begin{cases} \Delta u= 0\qquad &\text{in }\Omega,\\ u=g\qquad &\text{on }\partial\Omega\end{cases} \] attains the correct boundary values \(\lim_{x\to\xi} u(x)= g(x)\) as \(x\) approaches \(\xi\) in the domain \(\Omega\). This condition with a barrier is rather implicit. In 1924 N. Wiener gave a necessary and sufficient condition in terms of the electrostatic capacity of that part of the ball \(\overline B_r= \{x\in\mathbb{R}^n\mid|x-\xi|\leq r\}\) which lies in the complement of \(\Omega\). The celebrated Wiener criterion for regularity of the boundary point \(\xi\) is that the integral \[ \int_0 \frac{\text{cap}(\overline B_r\setminus\Omega)}{\text{cap}(\overline B_r)}\frac{dr}{r}=\infty \] diverges. The same Wiener criterion was extended to uniformly elliptic equations \[ \text{div}\langle{\mathbf A}(x),\nabla u\rangle=0 \] with bounded measurable coefficients by \textit{W. Littman} et al. [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 17, 43--77 (1963; Zbl 0116.30302)]. For a harmonic function, vanishing on the boundary near \(\xi\), Maz'ya obtained a \textit{modulus of continuity} in terms of the integral \[ \exp\Biggl[-\frac{n-2} {2(n-1)} \int^R_r \frac{\text{cap}(\overline B_r\setminus\Omega)}{\rho^{n-1}}\frac{d\rho}{\rho}\Biggr] \] appe…
reviewed by: Peter Lindqvist
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 31B30
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 31-02
zbMATH DE Number: 6949999
zbMATH Keywords: elliptic partial differential equations
zbMATH Keywords: Wiener criterion
zbMATH Keywords: boundary behaviour
zbMATH Keywords: irregular domains
author: V. Maz'ya
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/190
OpenAlex ID: W2892201713
description: scientific article; zbMATH DE number 6949999