*Accepted Paper*

**Inserted:** 2 dec 2013

**Last Updated:** 4 dec 2013

**Journal:** Discrete Contin. Dyn. Syst.

**Year:** 2013

**Abstract:**

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 $\nabla u$.

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