Global Lipschitz regularity in anisotropic elliptic problems with natural gradient growth

TL;DR

This paper proves global Lipschitz regularity for solutions to anisotropic elliptic boundary-value problems with natural gradient growth, under minimal assumptions on the data and domain geometry.

Contribution

It establishes the first regularity results for anisotropic p-Laplace type problems with natural growth conditions on the gradient.

Findings

Solutions are globally Lipschitz continuous.

Results hold for convex and minimally curved domains.

Weakest possible assumptions on the right-hand side are sufficient.

Abstract

We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. We establish global Lipschitz regularity of solutions under the weakest possible assumption on right-hand side of the equation, which may also include the gradient term with natural growth exponent. The results hold in either convex domains, or domains enjoying minimal integrability assumptions on the curvature of its boundary.