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

Abstract

We prove global $W^{1,q}(\Omega ,{\mathbb{R}}^N)$-regularity for minimisers of $\mathcal{F}(u)={\int }_{\Omega }F(x,\mathrm{D}u)\mathrm{d}x$ satisfying $u\ge \psi$ 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 $\alpha$-Hölder 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.

updated a small number of references. arXiv admin note: text overlap with arXiv:2010.15766

References in corpus (2)

• Lipschitz bounds and nonautonomous integrals
• Higher integrability for variational integrals with non-standard growth

Cited by in corpus (1)

• Regularity for multi-phase problems at nearly linear growth