Borderline gradient continuity of minima

Abstract

The gradient of any local minimiser of functionals of the type $$w\mapsto {\int }_{\Omega }f(x,w,Dw)dx+{\int }_{\Omega }w\mu dx,$$ where $f$ has $p$-growth, $p>1$, and $\Omega \subset {\mathbb{R}}^n$, is continuous provided the optimal Lorentz space condition $\mu \in L(n,1)$ is satisfied and $x\to f(x,\cdot )$ is suitably Dini-continuous.

30 pages