On the A-Obstacle Problem and the Hausdorff Measure of its Free Boundary

TL;DR

This paper establishes existence, uniqueness, and stability of solutions to the A-obstacle problem with L^1 data, and proves the free boundary has finite Hausdorff measure, extending previous results to more general operators.

Contribution

It introduces a framework for the A-obstacle problem with L^1 data, extending classical inequalities and analyzing the geometric measure of the free boundary.

Findings

Existence and uniqueness of entropy solutions for the A-obstacle problem.

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

Proof that the free boundary has finite Hausdorff measure.

Abstract

In this paper we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L^1 data. We also extend the Lewy-Stampacchia inequalities to the general framework of L^1 data, and show convergence and stability results. We then prove that the free boundary has finite N-1 Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p>2.