Radial Symmetry of Overdetermined Boundary-Value Problems in Exterior Domains |
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Authors: | Amandine Aftalion Jérôme Busca |
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Institution: | (1) DMI, Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France, FR |
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Abstract: | In this paper, we extend a classical result by J. Serrin 15] to exterior domains , where Ω is a bounded domain. We prove, under some hypotheses on f, that if there exists a solution of satisfying the overdetermined boundary conditions that and u are constant on , and such that , then the domain Ω is a ball. Under different assumptions on f, this result has been obtained by W. Reichel in 13]. The main result here covers new cases like with . When Ω is a ball, almost the same proof allows us to derive the symmetry of positive bounded solutions satisfying only the Dirichlet
condition that u is constant on . Our method relies on Kelvin transforms, various forms of the maximum principle and the device of moving planes up to a critical
position.
(Accepted May 30, 1997) |
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Keywords: | |
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