Symmetry of Nonnegative Solutions of Elliptic Equations via a Result of Serrin |
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Authors: | P. Polá[cbreve]ik |
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Affiliation: | 1. School of Mathematics , University of Minnesota , Minneapolis, Minnesota, USA polacik@math.umn.edu |
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Abstract: | We consider the Dirichlet problem for semilinear elliptic equations on a smooth bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction orthogonal to H. Employing Serrin's result on an overdetermined problem, we show that any nonzero nonnegative solution is necessarily strictly positive. One can thus apply a well-known result of Gidas, Ni and Nirenberg to conclude that the solution is reflectionally symmetric about H and decreasing away from the hyperplane in the orthogonal direction. |
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Keywords: | Elliptic equations Nonnegative solutions Symmetry |
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