Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems

Details Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems

2009-02-17

arxiv.org/abs/0902.2947v1

Mathematics

Optimization and Control

Scientific paper

We exhibit several counterexamples showing that the famous Serrin's symmetry result for semilinear elliptic overdetermined problems may not hold for partially overdetermined problems, that is when both Dirichlet and Neumann boundary conditions are prescribed only on part of the boundary. Our counterexamples enlighten subsequent positive symmetry results obtained by the first two authors for such partially overdetermined systems and justify their assumptions as well.

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