A Generalization of Stampacchia Lemma and Applications

TL;DR

This paper extends Stampacchia's Lemma to analyze the regularity of solutions to certain degenerate elliptic equations with variable coefficients, providing new tools for understanding solution behavior.

Contribution

It introduces a generalized version of Stampacchia Lemma applicable to degenerate elliptic equations with specific coefficient bounds, advancing regularity theory.

Findings

Established regularity results for weak and entropy solutions.

Derived bounds for solutions based on the generalized lemma.

Applied the theory to equations with coefficients depending on the solution.

Abstract

We present a generalization of Stampacchia Lemma and give applications to regularity property of weak and entropy solutions of degenerate elliptic equations of the form $$ \left\{ \begin{array}{llll} -\mbox{div} (a(x,u(x)) Du (x)) =f(x), & \mbox { in } \Omega, \\ u(x)=0, & \mbox { on } \partial \Omega, \end{array} \right. $$ where $$ \frac {\alpha}{(1+|u|) ^\theta} \le a(x,s)\le \beta $$ with $0<\alpha \le \beta <\infty$ and $0\le \theta <1$.