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

TL;DR

This paper investigates the properties of solutions to anisotropic, possibly singular or degenerate PDEs derived from a Wulff-type energy functional, establishing gradient bounds and symmetry results.

Contribution

It introduces new bounds on the gradient of solutions and derives rigidity and symmetry properties for solutions of anisotropic PDEs with singular or degenerate behavior.

Findings

Gradient of solutions is bounded by the potential F(u).

Rigidity and symmetry properties are established for solutions.

Results apply to anisotropic, possibly singular or degenerate PDEs.

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.