Niccolò Foralli

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

In this paper we prove a higher differentiability result for the solutions to a class of obstacle problems in the form \begin{equation*} \label{obst-def0} \min\left\{\int_ΩF(x,Dw) dx : w\in \mathcal{K}_ψ(Ω)\right\} \end{equation*} where $ψ\in W^{1,p(x)}(Ω)$ is a fixed function called obstacle and $\mathcal{K}_ψ=\{w \in W^{1,p(x)}_{0}(Ω)+u_0: w \ge ψ\,\, \textnormal{a.e. in $Ω$}\}$ 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 \begin{equation*}\label{growth}|ξ|^{p(x)}\le F(x,ξ)\le C(1+|ξ|^{p(x)}),\quad p(x)>1 \end{equation*}