Some constructions with solutions of variable coefficient elliptic equations
type: Item
instance of: scholarly article
title: Some constructions with solutions of variable coefficient elliptic equations
zbMATH Open document ID: 0802.31004
DOI: 10.1007/BF02921289
author: Thomas H. Wolff
published in: The Journal of Geometric Analysis
publication date: 1994-12-13T00:00:00Z
review text: The author considers the following inverse problem: Suppose that \(\Omega\) is a bounded domain in \(\mathbb{R}^ d\) with a smooth boundary \(\partial\Omega\), and \(f\) and \(g\) are smooth real-valued functions on \(\partial\Omega\). When does there exist a divergence-form elliptic equation div\((A\nabla u)=0\) with a solution \(u\) such that \(u|_{\partial\Omega}=f\), \(A\nabla u\cdot n|_{\partial\Omega}=g\)? (Here \(n\) is the exterior unit normal, \(A\) and \(u\) are supposed to be smooth.) Further, a characterization of such pairs of functions \(f\), \(g\) on the boundary of \(\Omega\) (more generally: on the boundary of a compact manifold) and some applications of related questions in potential theory are given.
Mathematics Subject Classification ID: 31B35
Mathematics Subject Classification ID: 31C12
zbMATH DE Number: 493651
zbMATH Keywords: critical point
zbMATH Keywords: divergence-form elliptic equation
reviewed by: Miroslav Brzezina
MaRDI profile type: Publication
cites work: Embedding of open Riemannian manifolds by harmonic functions
cites work: Nodal sets for solutions of elliptic equations
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full work available at URL: https://doi.org/10.1007/bf02921289
OpenAlex ID: W2074561366
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description: scientific article; zbMATH DE number 493651