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Global higher integrability for minimisers of convex obstacle problems with (p,q)-growth (2109.09485v2)

Published 20 Sep 2021 in math.AP

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.

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