On W1,q(⋅)-estimates for elliptic equations of p(x)-Laplacian type

Abstract

We study nonlinear elliptic equations of p(x)-Laplacian type on nonsmooth domains to obtain an optimal Calderón–Zygmund type estimate in the variable exponent spaces. We find a correct regularity assumption on p(⋅), a minimal regularity requirement on the associated nonlinearity and a suitable flatness condition on the boundary of the underlying domain for such W1,q(⋅) regularity theory to hold true for every variable exponent q(⋅) strictly bigger than p(⋅). Our result is a complete extension of the existing regularity results in the classical Lebesgue and Sobolev spaces from [14,25] to the one in the variable exponent space.