An excess-decay result for a class of degenerate elliptic equations
Maria Colombo, Alessio Figalli · Discrete and Continuous Dynamical Systems - S · 2014
We consider a family of degenerate elliptic equations of the form div $( abla F( abla u)) = f$, 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 $ abla u$ is close to a nondegenerate value for $F$. This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of $ abla u$ outside the degeneracy, and to anisotropic versions of the $p$-laplacian, where we get Hölder regularity of $ abla u$.