Regularity of solutions to the Dirichlet problem for degenerate elliptic equation
type: Item
instance of: scholarly article
title: Regularity of solutions to the Dirichlet problem for degenerate elliptic equation
zbMATH Open document ID: 1046.35019
DOI: 10.1142/S0252959903000530
author: Yemin Chen
published in: Chinese Annals of Mathematics. Series B
publication date: 2004-03-07T00:00:00Z
review text: The author considers the following Dirichlet problem \[ \begin{cases} Lu = - (a_{ij}u_{x_i})_{x_j} = f, &\text{ in }\Omega, \\ u=0, &\text{ on } \partial \Omega,\end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) and \(a_{ij}\) is a symmetric, measurable matrix such that \[ \exists \;\nu > 0 \;: \;\nu^{-1} \omega(x) | \xi| ^2 \leq a_{ij} \xi_i\xi_j \leq \nu \omega(x) | \xi| ^2\,, \;x \in \Omega\,, \;\xi \in \mathbb{R}^n\,, \] where \(\omega\) is a weight that belongs to the Muckenhoupt class \(A_2\). The aim of the paper is to describe how achieve regularity of a class of generalized solutions in terms of a suitable assumption on the known term \(f\). Essentially the paper contains the extension of some results already known in the uniformly elliptic case to the degenerate one. The results obtained are the following: Assume that the function \(f\) belongs to a class of potentials (Schechter class), then the author obtains an extra-integrability result. Under further assumptions on the function \(f\), the author proves that the corresponding solution coincides with the variational one, showing that it is more and more regular as the known term belongs to some Stummel-Kato class and Morrey classes. After that, his aim is to show that the conditions just proved are indeed also necessary. First he starts proving that all the conditions are necessary, assuming that the function \(f\) is nonnegative. After that, he tries to show that the same results hold true in case the function \(f\) has variable sign. Unfortunately, the paper contains a serious mistake because the results stated are false in the case of functions \(f\) with variable sign. The mistake is the following: the author tries to prove the variable sign case by reduction to the constant sign one by splitting the solution \(u\) in its positive and negative part. Then he states that, denoting by \(u^+\) and \(u^-\) respectively the positive and negative part of the solution, and setting \(f^+…
reviewed by: Giuseppe Di Fazio
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 2052052
zbMATH Keywords: degenerate elliptic equations
zbMATH Keywords: A2 weights
MaRDI profile type: Publication
full work available at URL: s0252959903000530
OpenAlex ID: W2321839977
description: scientific article; zbMATH DE number 2052052