Calculus of Variations and Geometric Measure Theory

M. Colombo - A. Figalli

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

created by colombom on 02 Dec 2013
modified on 04 Dec 2013


Accepted Paper

Inserted: 2 dec 2013
Last Updated: 4 dec 2013

Journal: Discrete Contin. Dyn. Syst.
Year: 2013


We consider a family of degenerate elliptic equations of the form ${\rm div} (\nabla F(\nabla u)) = f$, 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 H\"older regularity of $\nabla u$.