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A note on some overdetermined elliptic problem

Abstract : We define the notion of an exceptional manifold to be a flat Riemannian manifold with boundary which supports a positive harmonic function satisfying simultaneously a zero Dirichlet condition and a constant (nonzero) Neumann condtion at the boundary. We study the two-dimensional case: we present various examples and give a general construction algorithm of such surface by using complex analysis. We deduce a classification of all such surfaces assuming some further natural hypotheses and prove a Bernstein type theorem.
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https://hal.archives-ouvertes.fr/hal-00444828
Contributor : Frédéric Hélein <>
Submitted on : Thursday, January 7, 2010 - 12:44:46 PM
Last modification on : Friday, March 27, 2020 - 4:00:59 AM
Document(s) archivé(s) le : Friday, June 18, 2010 - 12:31:49 AM

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  • HAL Id : hal-00444828, version 1
  • ARXIV : 1001.1060

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Frédéric Hélein, Laurent Hauswirth, Frank Pacard. A note on some overdetermined elliptic problem. 2010. ⟨hal-00444828⟩

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