Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem
J. Andersson, G. S. Weiss
Abstract
We investigate singular and degenerate behavior of solutions of the unstable free boundary problem Δu = -χ\u>0\ . First, we construct a solution that is not of class C1,1 and whose free boundary consists of four arcs meeting in a 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.
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