Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem
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چکیده
We investigate singular and degenerate behavior of solutions of the unstable free boundary problem ∆u = −χ{u>0} . First, we construct a solution that is not of class C 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 the paper [6] by R. Monneau-G.S. Weiss. AMS Classification: 35R35, 35J60, 35B65.
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تاریخ انتشار 2008