Optimal gradient estimates via Riesz potentials for p(·)-Laplacian type equations

Abstract

We investigate degenerate elliptic equations with coefficients of p(center dot)-Laplacian type when the right-hand side is a finite Borel measure. An optimal pointwise estimate of the gradient of a very weak solution to such a measure data problem is obtained via a Riesz potential of the measure under a minimal regularity requirement on the associated coefficients and an optimal condition on the variable exponent p(center dot). As a consequence, we are able to derive a C-1 regularity criterion as well as a Calderon-Zygmund-type estimate in the literature.