Analysis on the junctions of domain walls

Luis  A. Caffarelli; Fang Hua Lin

doi:10.3934/dcds.2010.28.915

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2010, Volume 28, Issue 3: 915-929. Doi: 10.3934/dcds.2010.28.915

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Analysis on the junctions of domain walls

Luis A. Caffarelli^1, and
Fang Hua Lin^2,

1.
University of Texas at Austin, Department of Mathematics, 1 University Station, C1200, Austin, TX 78712-1082
2.
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185

Received: March 2010

Revised: April 2010

Published: September 2010

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we study the local structure and the smoothness of singularities of free boundaries in an optimal partition problem for the Dirichlet eigenvalues. We prove that there is a unique homogeneous blow up(tangent map) at each singular point in the interior of the free boundary. As a consequence we obtain the rectifiability as well as local structures of singularities.

Keywords:

free boundaries; calculus of variations; montonicity formula.

Mathematics Subject Classification: 35R35; 49R99.

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