Gradient continuity for p(x)-Laplacian systems under minimal conditions on the exponent

Abstract

We consider solutions of p(x)-Laplacian systems with coefficients and we show that their gradient is continuous provided that the variable exponent has distributional gradient belonging to the Lorentz-Zygmund space Ln,1log⁡L and that the gradient of the coefficient belongs to the Lorentz space Ln,1. The result is new since the use of the sharp Sobolev embedding in rearrangement invariant spaces does not ensure the unique (up to now) known assumption for such result, namely the log-Dini continuity of p(⋅) and the plain Dini continuity of the coefficient. Our approach relies on perturbation arguments and allows to slightly improve results in dimension two even for the case where p(⋅) is constant.