Regularity results for bounded solutions to obstacle problems with non-standard growth conditions

Abstract

In this paper we consider a class of obstacle problems of the type %\begin{equation*} %\int_Ω\left<A(x, Du), D(φ-u)\right> \, \dx\ge0\qquad\forall %φ\in W^{1,q}(Ω) \quad {\mathrm{s.t.}} \quad φ\ge ψ%\end{equation*} \begin{equation*} \min \left\{\int_Ωf(x, Dv)\, \dx\,:\, v\in \mathcal{K}_ψ(Ω)\right\} \end{equation*} where $\psi$ is the obstacle, ${\mathcal{K}}_{\psi }(\Omega )=\{v\in u_0+W_0^{1,p}(\Omega ,\mathbb{R}):v\ge \psi \text{ a.e. in }\Omega \}$, with $u_0\in W^{1,p}(\Omega )$ a fixed boundary datum, the class of the admissible functions and the integrand $f(x,Dv)$ satisfies non standard $(p,q)$-growth conditions. \\ We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map $x\mapsto A(x,\xi )$ is independent of the dimension $n$ and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one $W^{1,n}$.