An excess-decay result for a class of degenerate elliptic equations

Maria Colombo,Alessio Figalli

doi:10.3934/dcdss.2014.7.631

Abstract

We consider a family of degenerate elliptic equations of the form div $(\nabla F(\nabla u)) = f$, where $F\in C^{1,1}$ is a convex function which is elliptic outside a ball.We prove an excess-decay estimate at points where $\nabla u$ is close to a nondegenerate value for $F$. This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of $\nabla u$ outside the degeneracy, and to anisotropic versions of the $p$-laplacian, where we get Hölder regularity of $\nabla u$.

Highlights

We study the local regularity of minimizers of the functional
Even in the vectorial case, partial regularity of minimizers was proved under the uniform strict quasiconvexity assumption in [13, 1]
To understand regularity for more degenerate elliptic problems, a natural idea is to prove Holder regularity at points where the gradient is close to a value where the function F is C2 and uniformly convex

Summary

Introduction

Ω where Ω ⊆ Rn is an open set, F : Rn → R, f : Ω → R, and u : Ω → R. To understand regularity for more degenerate elliptic problems, a natural idea is to prove Holder regularity at points where the gradient is close to a value where the function F is C2 and uniformly convex. This scheme has been carried out by Anzellotti and Giaquinta in [3] under the uniform convexity assumption for elliptic systems and in [2] if uniform strict quasiconvexity is assumed. For some ξ0 ∈ RnN and x0 ∈ Rn, F is C2 in a neighborhood of ξ0, and a uniform strict quasiconvexity holds true around ξ0, u is of class C1,α in a neighborhood of x0 for every α < 1 Their proof is based on a linearization argument.

Main result

Applications

Proofs