The Dirichlet problem with \(L^ 2\)-boundary data for elliptic linear equations
type: Item
instance of: scholarly article
title: The Dirichlet problem with \(L^ 2\)-boundary data for elliptic linear equations
zbMATH Open document ID: 0734.35024
published in: Lecture Notes in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: It is considered the Dirichlet problem with \(L^ 2\)-boundary data for linear second order elliptic equations of the form \[ -D_ i(a^{ij}(x)D_ ju(x)+b^ i(x)u(x))+d^ i(x)D_ iu(x)+c(x)u(x)=f(x)\text{ in } Q, \] \[ u(x)=\phi (x)\text{ on } \partial Q \] where Q is a (sufficiently regular) bounded domain in \(R_ n\). Here \(\phi\) is assumed to be in \(L^ 2(\partial Q)\), therefore we cannot expect to find a solution in the space \(W^{1,2}(Q)\), since not every function in \(L^ 2(\partial Q)\) is the trace of a function of \(W^{1,2}(Q)\). This difficulty is overcome by introducing a suitable weighted Sobolev space \(\tilde W^{1,2}(Q)\) instead of \(W^{1,2}(Q)\), and by considering traces of the solution on surfaces parallel to \(\partial Q.\) It would be too long to list all the results of this work, which involves also reverse Hölder inequalities, harmonic measures, degenerate ellipticity, Carleson measures. The hypotheses on the boundary \(\partial Q\) and on the coefficients \(a^{ij}\) are often more general than in previous papers on the same subject. In conclusion, the book is almost self contained, well written, up-to- date and it will certainly interest the numerous specialists on elliptic equations.
reviewed by: Maurizio Chicco
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 51253
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: \(L^ 2\)-boundary data
zbMATH Keywords: weighted Sobolev space
zbMATH Keywords: traces
zbMATH Keywords: reverse Hölder inequalities
zbMATH Keywords: harmonic measures
zbMATH Keywords: Carleson measures
author: Jan Chabrowski
MaRDI profile type: Publication
DOI: 10.1007/BFB0095750
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description: scientific article; zbMATH DE number 51253