Regularity results to the class of variational obstacle problems with variable exponent

Abstract

In this paper we prove local gradient estimates and higher differentiability result for the solutions of variational obstacle inequalities \int_\Omega\big<\mathcal{A}(x,u,Du),D(\phi-u)\big>dx\geq \int_\Omega\mathcal{B}(x,u,Du)(\phi-u)dx. for all $\phi\in \mathcal{K}_\psi(\Omega)$. Here $\Omega(\subset\mathbb{R}^n)$ is bounded, $n\geq 2$ and $\psi:\Omega\rightarrow\mathbb{R}$ is called obstacle. Here we deal with variable exponent growth , namely $p(.)$-growth . At first we prove Calder\'on-Zygmund estiamte and then using this result to prove higher differentiability result in Besov scale.