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.
Categories: math.AP