*Accepted Paper*

**Inserted:** 11 oct 2012

**Last Updated:** 16 apr 2013

**Journal:** J. Math. Pures Appl.

**Year:** 2012

**Abstract:**

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

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