A Rigidity Result for Overdetermined Elliptic Problems in the Plane

By A. Ros, D. Ruiz, P. Sicbaldi. 2015

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Let f:[0,+∞)→ℝ be a (locally) Lipschitz function and Ω⊂ℝ2 a C1,α domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem [{ Δu+f(u)=0in Ω,u=0on Ω,∂u∂ν→=1on 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.© 2017 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

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