A Rigidity Result for Overdetermined Elliptic Problems in the Plane. Issue 7 (22nd April 2017)

• Record Type: Journal Article
• Authors: Ros, Antonio; Ruiz, David; Sicbaldi, Pieralberto
• Is Part Of: Communications on pure and applied mathematics. Volume 70:Issue 7(2017:Jul.)
• Page Start: 1223
• Page End: 1252
• Publication Date: 2017-04-22
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21696
• Languages: English
• ISSNs: 0010-3640

Abstract

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 [ { Δ 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.© 2017 Wiley Periodicals, Inc.