A Hölder continuity result for a class of obstacle problems under non standard growth conditions

We prove C 0, α regularity for local minimizers u of functionals with p ( x )‐growth of the type\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ {\mathcal F}(w,\Omega ) := \int _\Omega f(x,w(x),Dw(x))\, dx, $$ \end{document} in the class \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$K :=\lbrace w \in W^{1,p(\cdot )}(\Omega ;{\mathbb R}): w \ge \psi \rbrace$\end{document} , where the exponent function p : Ω → (1, ∞) is assumed to be continuous with a modulus of continuity satisfying\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ \limsup _{\rho \rightarrow 0} \omega (\rho )\log \left(\frac{1}{\rho }\right) < +\infty , $$ \end{document} and 1 < γ 1 ⩽ p ( x ) ≤ γ 2 < +∞. Moreover, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\psi \in W^{1,1}_{\textnormal {loc}}(\Omega )$\end{document} is a given obstacle function, whose gradient D ψ belongs to a Morrey space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L_{\textnormal {loc}}^{q,\lambda }(\Omega )$\end{document} with n − γ 1 < λ < n and q > γ 2 . We do not assume any quantitative continuity of the integrand function f .