On the continuity of solutions to degenerate elliptic equations
type: Item
instance of: scholarly article
title: On the continuity of solutions to degenerate elliptic equations
zbMATH Open document ID: 1213.35231
published in: Journal of Differential Equations
publication date: 2011-03-14T00:00:00Z
review text: The very interesting paper under review deals with the local behaviour of weak solutions to the divergence form equation \[ \text{div\,} A(x)\nabla u=0\qquad \text{in}\;\Omega\subset\mathbb R^n \] with a real and symmetric matrix \(A(x)\) of measurable entries satisfying \[ w(x)|\xi|^2\leq \big\langle A(x)\xi,\xi\big\rangle\leq v(x)|\xi|^2 \] with suitable weights \(w(x)\geq0\) and \(v(x)\). In the special case of \(A_2\)-Muckenhoupt weights satisfying \(v(x)=\Lambda w(x)\) a.e. in \(\Omega\) with \(\Lambda\geq1,\) \textit{S. Chanillo} and \textit{R. L. Wheeden} [Commun. Partial Differ. Equations 11, 1111--1134 (1986; Zbl 0634.35035)] proved continuity of the weak solution under additional, very strong assumptions on the oscillations \(w(x)\) and \(v(x)\). In the paper under review, the authors consider the case \(w(x) = 1/K(x)\) and \(v(x) = K(x)\in A_2.\) Assuming \(K\) to satisfy the Gehring \(G_n\)-condition \[ G_n(K)=\sup_B { {\left({1\over{|B|}}\int_B K(x)^ndx\right)^{1/n}} \over{{1\over{|B|}}\int_B K(x)dx} }<\infty \] with the supremum taken over all balls \(B\subset \mathbb R^n,\) and setting \(S_v\) for the domain of the maximal function of \(v\), \(S_v =\{x\in\Omega:\;Mv(x)<\infty\}\), it is proved that the restriction to \(S_v\) of the precise representative \(\tilde u\) of any non-negative solution \(u\) is continuous.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 35D30
zbMATH DE Number: 5865665
zbMATH Keywords: partial differential equation
zbMATH Keywords: divergence form
zbMATH Keywords: Muckenhoupt weight
zbMATH Keywords: finite distortion
zbMATH Keywords: maximal function
zbMATH Keywords: Gehring class
author: D. Cruz-Uribe
author: P. Di Gironimo
author: Carlo Sbordone
MaRDI profile type: Publication
full work available at URL: j.jde.2010.12.014
OpenAlex ID: W1998126646
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DOI: 10.1016/J.JDE.2010.12.014
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description: scientific article; zbMATH DE number 5865665