The obstacle problem for nonlinear elliptic equations with variable growth and L^1-data

TL;DR

This paper establishes existence, uniqueness, and stability of entropy solutions for obstacle problems involving nonlinear elliptic equations with variable growth and L^1-data, extending classical inequalities to this setting.

Contribution

It introduces a framework for solving obstacle problems with variable growth and L^1-data, proving key properties and extending Lewy-Stampacchia inequalities.

Findings

Existence and uniqueness of entropy solutions for the problem.

Convergence and stability of the coincidence set.

Extension of Lewy-Stampacchia inequalities to L^1-data.

Abstract

The aim of this paper is twofold: to prove, for L^1-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy--Stampacchia inequalities to the general framework of L^1.