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

TL;DR

This paper establishes higher differentiability of solutions to obstacle problems with variable exponent growth conditions, extending regularity results in the calculus of variations for non-standard growth models.

Contribution

It proves higher differentiability for solutions to obstacle problems involving variable exponent growth, a novel extension in the regularity theory for such problems.

Findings

Solutions exhibit higher differentiability under variable exponent conditions.

The results apply to convex integrands with p(x)-growth.

Enhanced regularity results for obstacle problems with non-uniform growth.

Abstract

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_\Omega F(x,Dw) dx : w\in \mathcal{K}_{\psi}(\Omega)\right\} \end{equation*} where $\psi\in W^{1,p(x)}(\Omega)$ is a fixed function called obstacle and $\mathcal{K}_{\psi}=\{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 \begin{equation*}\label{growth}|\xi|^{p(x)}\le F(x,\xi)\le C(1+|\xi|^{p(x)}),\quad p(x)>1 \end{equation*}