Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations

Abstract

We consider the Wulff-type energy functional $${\mathcal{W}}_{\Omega }(u):={\int }_{\Omega }B(H(\nabla u(x)))-F(u(x))dx,$$ where $B$ is positive, monotone and convex, and $H$ is positive homogeneous of degree 1. The critical points of this functional satisfy a possibly singular or degenerate, quasilinear equation in an anisotropic medium. We prove that the gradient of the solution is bounded at any point by the potential $F(u)$ and we deduce several rigidity and symmetry properties.

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