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On an overdetermined
elliptic problem
Laurent Hauswirth, Frédéric Hélein and Frank Pacard |
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Vol. 250 (2011), No. 2, 319–334
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Abstract |
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A smooth flat Riemannian manifold is called an exceptional domain if it admits positive harmonic functions having vanishing Dirichlet boundary data and constant (nonzero) Neumann boundary data. In analogy with minimal surfaces, a representation formula is derived and applied to the classification of exceptional domains. Some interesting open problems are proposed along the way. |
Keywords
harmonic function, overdetermined elliptic problem,
extremal domain, minimal surface
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Mathematical Subject Classification 2000
Primary: 35N25, 35J25, 35R35
Secondary: 53A10
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Milestones
Received: 4 January 2010
Accepted: 12 April 2010
Published: 31 March 2011
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Authors |
| Laurent Hauswirth | |
| Université Paris-Est
Marne-la-Vallée Laboratoire d’Analyse et Mathématiques Appliquées Cité Descartes 5 Boulevard Descartes Champs-sur-Marne 77454 Marne-la-Vallée Cedex 2 France |
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| http://perso-math.univ-mlv.fr/users/hauswirth.laurent | |
| Frédéric Hélein | |
| Université Paris Diderot–Paris
7 Institut de Mathématiques de Jussieu, UMR CNRS 7586 UFR de Mathématiques, Case 7012 Bâtiment Chevaleret 75205 Paris Cedex 13 France |
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| http://www.math.jussieu.fr/~helein/ | |
| Frank Pacard | |
| Institut Universitaire de France et
Centre de Mathématiques Laurent Schwartz UMR CNRS 7640 École Polytechnique 91128 Palaiseau France |
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| http://www.math.polytechnique.fr/~pacard | |
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