  We prove global $W^{1,q}(\Omega,\mathbb{R}^m)$-regularity for minimisers of
convex 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 of
the associated relaxed functional. Our main assumptions on $F(x,z)$ are a
uniform $\alpha$-H\"older continuity assumption in $x$ and controlled
$(p,q)$-growth conditions in $z$ with $q<\frac{(n+\alpha)p}{n}$.
