An excess-decay result for a class of degenerate elliptic equations

Maria Colombo; Alessio Figalli

doi:10.3934/dcdss.2014.7.631

Article Contents

2014, Volume 7, Issue 4: 631-652. Doi: 10.3934/dcdss.2014.7.631 open access

This issue Previous Article Some degenerate parabolic problems: Existence and decay properties Next Article Hardy-Littlewood-Sobolev systems and related Liouville theorems

An excess-decay result for a class of degenerate elliptic equations

Maria Colombo^1, and
Alessio Figalli^2,

1.
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa
2.
University of Texas at Austin, Department of Mathematics, 2515 Speedway Stop C1200, Austin, TX 78712-1202

Received: September 30, 2013

Revised: November 30, 2013

Published: July 2014

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Abstract

We consider a family of degenerate elliptic equations of the form 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 Hölder regularity of $\nabla u$.

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Mathematics Subject Classification: Primary: 35J70; Secondary: 35B65, 49K20.

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