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29 March 2024
 
  » arxiv » math.AP/0508184

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Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem
J. Andersson ; G.S. Weiss ;
Date 10 Aug 2005
Subject Analysis of PDEs MSC-class: 35R35; 35J60 | math.AP
AbstractWe investigate singular and degenerate behavior of solutions of the unstable free boundary problem $$Delta u = -chi_{{u>0}} .$$ First, we construct a solution that is not of class $C^{1,1}$ and whose free boundary consists of four arcs meeting in a {em cross}-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is non-trivial. We also show existence of a degenerate solution. This answers two of the open questions in a recent paper by R. Monneau-G.S. Weiss.
Source arXiv, math.AP/0508184
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