We prove global W1,q(Ω,RN)\documentclass[12pt]{minimal}
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\begin{document}$$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$\end{document}-regularity for minimisers of F(u)=∫ΩF(x,Du)dx\documentclass[12pt]{minimal}
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\begin{document}$${{\mathscr {F}}(u)=\int _\Omega F(x,Du)\,{\mathrm{d}}x}$$\end{document} satisfying u≥ψ\documentclass[12pt]{minimal}
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\begin{document}$$u\ge \psi $$\end{document} for a given Sobolev obstacle ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}. W1,q(Ω,RN)\documentclass[12pt]{minimal}
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\begin{document}$$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$\end{document} regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}-Hölder continuity assumption in x and natural (p, q)-growth conditions in z with q<(n+α)pn\documentclass[12pt]{minimal}
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\begin{document}$$q<\frac{(n+\alpha )p}{n}$$\end{document}. In the autonomous case F≡F(z)\documentclass[12pt]{minimal}
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\begin{document}$$F\equiv F(z)$$\end{document} we can improve the gap to q<minnpn-1,p+1\documentclass[12pt]{minimal}
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\begin{document}$$q<\min \left( \frac{np}{n-1},p+1\right) $$\end{document}, a result new even in the unconstrained case.
