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Journal Articles Communications on Pure and Applied Mathematics Year : 2017

A RIGIDITY RESULT FOR OVERDETERMINED ELLIPTIC PROBLEMS IN THE PLANE

Abstract

Let f : [0, +∞) → R be a (locally) Lipschitz function and Ω ⊂ R 2 a C 1,α domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem        ∆u + f (u) = 0 in Ω u = 0 on ∂Ω ∂u ∂ ν = 1 on ∂Ω we prove that Ω is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997.
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Dates and versions

hal-01481862 , version 1 (03-03-2017)
hal-01481862 , version 2 (11-11-2017)

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Antonio Ros, David Ruiz, Pieralberto Sicbaldi. A RIGIDITY RESULT FOR OVERDETERMINED ELLIPTIC PROBLEMS IN THE PLANE. Communications on Pure and Applied Mathematics, 2017, 70 (7), pp.1223 - 1252. ⟨10.1002/cpa.21696⟩. ⟨hal-01481862v2⟩
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