Regularity for Obstacle Problems without Structure Conditions

TL;DR

This paper establishes Lipschitz regularity for minimizers in a broad class of obstacle problems without relying on traditional structure conditions, using relaxed functional techniques and a key lemma.

Contribution

It introduces a novel approach leveraging relaxed functionals and a crucial lemma to prove regularity without standard assumptions.

Findings

Proves Lipschitz regularity of minimizers in obstacle problems

Handles problems with Lavrentiev phenomenon without structure conditions

Utilizes relaxed functional framework and a key lemma

Abstract

This paper deals with the Lipschitz regularity of minimizers for a class of variational obstacle problems with possible occurance of the Lavrentiev phenomenon. In order to overcome this problem, the availment of the notions of relaxed functional and Lavrentiev gap are needed. The main tool used here is a ingenious Lemma which reveals to be crucial because it allows us to move from the variational obstacle problem to the relaxed-functional-related one. This is fundamental in order to find the solutions' regularity that we intended to study. We assume the same Sobolev regularity both for the gradient of the obstacle and for the coefficients.