Overdetermined problems with possibly degenerate ellipticity, a geometric approach |
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Authors: | Ilaria Fragalà Filippo Gazzola Bernd Kawohl |
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Institution: | 1. Dipartimento di Matematica, Politecnico, Piazza L. da Vinci 20133, Milano, Italy 2. Mathematisches Institut, Universit?t K?ln, 50923, K?ln, Germany
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Abstract: | Given an open bounded connected subset Ω of ℝn, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary
data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under
fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle
for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω. |
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Keywords: | Overdetermined boundary value problem Degenerate elliptic operators |
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