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

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