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July  2010, 28(3): 915-929. doi: 10.3934/dcds.2010.28.915

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, United States
2. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, United States

Received  March 2010 Revised  April 2010 Published  April 2010

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; 49R9.
Citation: Luis A. Caffarelli, Fang Hua Lin. Analysis on the junctions of domain walls. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 915-929. doi: 10.3934/dcds.2010.28.915
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