Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem

Abstract

We 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.

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• Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes
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