The obstacle problem for nonlinear degenerate equations with $L^{1}$-data

TL;DR

This paper investigates the obstacle problem for nonlinear degenerate elliptic equations with $L^{1}$ data, establishing existence of entropy solutions and their regularity in Sobolev spaces.

Contribution

It introduces new existence results for entropy solutions to degenerate elliptic obstacle problems with $L^{1}$ data and proves their Sobolev regularity.

Findings

Existence of entropy solutions under $L^{1}$ data assumptions.

Entropy solutions belong to certain Sobolev spaces.

The results extend the understanding of degenerate elliptic obstacle problems.

Abstract

The aim of this paper is to study the obstacle problem with an elliptic operator having degenerate coercivity. We prove the existence of an entropy solution to the obstacle problem under the assumption of $L^{1}-$summability on the data. Meanwhile, we prove that every entropy solution belongs to some Sobolev space $W^{1,q}(\Omega)$.