# Regularity results for a very degenerate elliptic equation with applications to traffic dynamic

##
Maria Colombo
(University of Zurich / ETH)

created by magnani on 28 Nov 2012

modified by colombom on 17 Apr 2013

5 dec 2012
-- 17:00
[open in google calendar]

Sala Seminari, Department od Mathematics, Pisa University

**Abstract.**

We consider the equation
$$\nabla \cdot (\nabla F(\nabla u)) = f,$$
where $F:{\mathbb R}^n\to{\mathbb R}$ is a convex function. We are interested in the regularity of solutions in the case of a strongly degenerate $F$. In particular, we consider a function $F$ which is $0$ inside some convex set $E$ and is uniformly elliptic outside $E$. The particular choice of $F$ arises in the context of traffic congestion.

Under some natural assumptions on $F$ and $f$, we prove that the function $\nabla F(\nabla u)$ is continuous in any dimension, extending a previous result by Santambrogio and Vespri valid only in dimension $2$.

(joint work with Alessio Figalli)