Quasiconvexity and partial regularity via nonlinear potentials

TL;DR

This paper develops sharp partial regularity results for relaxed minimizers of degenerate quasiconvex functionals with $(p,q)$-growth, utilizing nonlinear potential theory to establish optimal local regularity criteria under minimal assumptions.

Contribution

It introduces a novel approach using nonlinear potential theory to derive optimal regularity criteria for quasiconvex functionals with $(p,q)$-growth, advancing the understanding of partial regularity in degenerate elliptic problems.

Findings

Established sharp partial regularity results for relaxed minimizers.

Derived optimal local regularity criteria under minimal assumptions.

Applied nonlinear potential theory to quasiconvex functionals with $(p,q)$-growth.

Abstract

We show how to infer sharp partial regularity results for relaxed minimizers of degenerate, nonuniformly elliptic quasiconvex functionals, using tools from Nonlinear Potential Theory. In particular, in the setting of functionals with $(p,q)$-growth - according to the terminology of Marcellini [52] - we derive optimal local regularity criteria under minimal assumptions on the data.