Calculus of Variations and Geometric Measure Theory

M. Colombo - A. Figalli

Regularity results for very degenerate elliptic equations

created by colombom on 11 Oct 2012
modified on 16 Apr 2013


Accepted Paper

Inserted: 11 oct 2012
Last Updated: 16 apr 2013

Journal: J. Math. Pures Appl.
Year: 2012


We consider a family of elliptic equations introduced in the context of traffic congestion. They have the form $\nabla \cdot (\nabla \mathcal{F}(\nabla u)) = f$, where $\mathcal{F}$ is a convex function which vanishes inside some convex set and is elliptic outside. Under some natural assumptions on $\mathcal{F}$ and $f$, we prove that the function $\nabla \mathcal{F}(\nabla u)$ is continuous in any dimension, extending a previous result by Santambrogio and Vespri valid only in dimension $2$.

Keywords: regularity, Degenerate elliptic PDEs