Higher differentiability of solutions for a class of obstacle problems with variable exponents

Abstract

In this paper we prove a higher differentiability result for the solutions to a class of obstacle problems in the form
\n \\begin{equation*}
\n\\label{obst-def0}
\n\\min\\left\\{\\int_\\Omega F(x,Dw) dx : w\\in \\mathcal{K}_{\\psi}(\\Omega)\\right\\}
\n\\end{equation*}
\nwhere $\\psi\\in W^{1,p(x)}(\\Omega)$ is a fixed function called obstacle and $\\mathcal{K}_{\\psi}(\\Omega)=\\{w \\in W^{1,p(x)}_{0}(\\Omega)+u_0: w \\ge \\psi \\,\\, \\textnormal{a.e. in $\\Omega$}\\}$ is the class of the admissible functions, for a suitable boundary value $ u_0 $. We deal with a convex integrand $F$ which satisfies the $p(x)$-growth conditions
\n\\begin{equation*}\\label{growth}|\\xi|^{p(x)}\\le F(x,\\xi)\\le C(1+|\\xi|^{p(x)}),\\quad p(x)>1
\n\\end{equation*}

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