Lipschitz regularity for solutions of a general class of elliptic equations

Abstract

We prove local Lipschitz regularity for local minimiser of \[ W^{1,1}(\Omega)\ni v\mapsto \int_\Omega F(Dv)\, dx \] where $\Omega\subseteq {\mathbb R}^N$, $N\ge 2$ and $F:{\mathbb R}^N\to {\mathbb R}$ is a quasiuniformly convex integrand in the sense of Kovalev and Maldonado, i.e. a convex $C^1$-function such that the ratio between the maximum and minimum eigenvalues of $D^2F$ is essentially bounded. This class of integrands inculdes the standard singular/degenerate functions $F(z)=|z|^p$ for any $p>1$ and arises naturally as the closure, with respect to a natural convergence, of the strongly elliptic integrands of the Calculus of Variations.