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

Abstract

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

31 pages, v2 with minor changes to the introduction and typos corrected

References in corpus (1)

• Higher integrability for variational integrals with non-standard growth