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A Rigidity Result for Overdetermined Elliptic Problems in the Plane
Author(s) -
Ros Antonio,
Ruiz David,
Sicbaldi Pieralberto
Publication year - 2017
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21696
Subject(s) - overdetermined system , bounded function , mathematics , rigidity (electromagnetism) , plane (geometry) , domain (mathematical analysis) , lipschitz continuity , boundary (topology) , pure mathematics , mathematical analysis , elliptic curve , geometry , physics , quantum mechanics
Let f : [ 0 , + ∞ ) → ℝ be a (locally) Lipschitz function and Ω ⊂ ℝ 2aC 1 , αdomain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem[ {Δ u + 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.