Gradient regularity for a class of elliptic obstacle problems

TL;DR

This paper establishes regularity results, including higher differentiability and Lipschitz continuity, for local minimizers of obstacle problems involving non-autonomous integral functionals with Sobolev regular coefficients.

Contribution

It introduces new regularity results for obstacle problems with Sobolev regular coefficients, extending previous understanding of minimizer smoothness.

Findings

Higher differentiability of minimizers

Lipschitz continuity of solutions

Regularity results under Sobolev coefficient assumptions

Abstract

We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,\Omega)=\int_\Omega F(x,Dv)dx,$$ under the constraint $v \ge \psi$ a.e. in $\Omega$, where $\psi$ is a fixed obstacle function. Assuming that the coefficients of the partial map $x \mapsto D_\xi F(x,\xi)$ satisfy a suitable Sobolev regularity, we are able to obtain higher differentiability and Lipschitz continuity results for the local minimizers.