Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems

Ilaria Fragalà (1) , Filippo Gazzola (1) , Jimmy Lamboley (2) , Michel Pierre (2)

1 Department of mathematics Francesco Brioschi
2 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: overdetermined boundary value problems; shape optimization

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Ilaria Fragalà, Filippo Gazzola, Jimmy Lamboley, Michel Pierre. Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems. Analysis, 2009, 29 (1), pp.85-93. ⟨hal-00362312⟩