A boundary regularity result for minimizers of variational integrals with nonstandard growth

TL;DR

This paper establishes global Lipschitz regularity for minimizers of a broad class of convex variational integrals without geometric constraints or domain restrictions, using novel approximation and barrier techniques.

Contribution

It introduces a new approach to prove regularity for variational integrals with nonstandard growth, removing previous geometric and domain assumptions.

Findings

Proves Lipschitz regularity for minimizers in general domains.

Develops a new approximation method for the functional.

Constructs barrier functions to facilitate regularity proof.

Abstract

We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in $W^{1,1}$ with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as for example bounded slope condition). Furthermore, we do not assume any restrictive assumption on the geometry of the domain and the result is valid for all sufficiently smooth domains. The result is achieved with a suitable approximation of the functional together with a new construction of appropriate barrier functions.