Calculus of Variations and Geometric Measure Theory
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Overdetermined elliptic problems and a conjecture of Berestycki, Caffarelli and Nirenberg

David Ruiz

created by malchiodi on 17 Jul 2017

25 jul 2017 -- 14:00   [open in google calendar]

Scuola Normale Superiore, Aula Bianchi


n this talk we consider an elliptic semilinear problem under overdetermined boundary conditions: the solution vanishes at the boundary and the normal derivative is constant. These problems appear in many contexts, as in the study of free boundaries and obstacle problems. Here the task is to understand for which domains (called extremal domains) we may have a solution. This question has shown a certain parallelism with the theory of CMC surfaces, and also with the well-known De Giorgi conjecture for the Allen-Cahn equation.

The case of bounded extremal domains was completely solved by J. Serrin in 1971, and the ball is the unique such domain. Instead, the case of unbounded domains is far from being completely understood. In this talk we give a rigidity result in dimension 2, and also a construction of a nontrivial extremal domain in the form of a exterior domain.

‚ÄčThis is joint work with Antonio Ros (U. Granada) and Pieralberto Sicbaldi (U. Granada and U. Aix Marseille).‚Äč

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