Global higher integrability for minimisers of convex functionals with (p,q)-growth

TL;DR

This paper establishes global higher integrability (W^{1,q}) regularity for minimizers of convex functionals with controlled (p,q)-growth, under specific regularity and growth conditions, extending regularity results in calculus of variations.

Contribution

It proves the first global W^{1,q} regularity results for minimizers of convex functionals with (p,q)-growth under Hölder continuity in the spatial variable.

Findings

Global W^{1,q} regularity for minimizers

Regularity for relaxed functional minimizers

Conditions on (p,q)-growth and Hölder continuity

Abstract

We prove global $W^{1,q}(\Omega,\mathbb{R}^m)$-regularity for minimisers of convex functionals of the form $\mathscr{F}(u)=\int_\Omega F(x,Du)\mathrm{d} x$. $W^{1,q}(\Omega,\mathbb{R}^m)$ regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on $F(x,z)$ are a uniform $\alpha$-H\"older continuity assumption in $x$ and controlled $(p,q)$-growth conditions in $z$ with $q<\frac{(n+\alpha)p}{n}$.