Lipschitz regularity for solutions of a general class of elliptic equations

Abstract

We prove local Lipschitz regularity for local minimiser of \[ W^{1,1}(Ω)\ni v\mapsto \int_ΩF(Dv)\, dx \] 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 $D^{2}F$ 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.

typo fixed in (1.5), 37 pages, comments welcome