Besov Regularity Estimates for a Class of Obstacle Problems with Variable Exponents

Abstract

In this paper we obtain the local regularity estimates in Besov spaces of weak solutions for a class of elliptic obstacle problems with variable exponents \(p(x)\). We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality in the following form

$$\begin{aligned} \int _{\Omega } \langle A\left (x, Du \right ),~D \left (\varphi -u \right )\rangle {\mathrm{d}}x\geq \int _{\Omega } \langle F,~D \left ( \varphi -u \right )\rangle {\mathrm{d}}x \end{aligned}$$

under some proper assumptions on the function \(p(x)\), \(A\), \(\varphi \) and \(F\). Moreover, we would like to point out that our results improve the known results for such problems.