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

TL;DR

This paper establishes global higher integrability for minimisers of convex obstacle problems with (p,q)-growth, extending regularity results under specific growth and continuity conditions, including new bounds in the autonomous case.

Contribution

It proves global $W^{1,q}$-regularity for obstacle problem minimisers under (p,q)-growth, with improved bounds in the autonomous case, advancing regularity theory for such variational problems.

Findings

Global $W^{1,q}$-regularity for obstacle minimisers

Improved bounds in autonomous case to $q<\frac{np}{n-1}$

Regularity results under $(p,q)$-growth and Hölder continuity assumptions

Abstract

We prove global $W^{1,q}(\Omega,\mathbb{R}^N)$-regularity for minimisers of $\mathscr{F}(u)=\int_\Omega F(x,\mathrm{D}u)\mathrm{d} x$ satisfying $u\geq \psi$ for a given Sobolev obstacle $\psi$. $W^{1,q}(\Omega,\mathbb{R}^m)$ 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 natural $(p,q)$-growth conditions in $z$ with $q<\frac{(n+\alpha)p}{n}$. In the autonomous case $F\equiv F(z)$ we can improve the gap to $q<\frac{np}{n-1}$, a result new even in the unconstrained case.