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Overdetermined problems with possibly degenerate ellipticity, a geometric approach

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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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Correspondence to Bernd Kawohl.

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Fragalà, I., Gazzola, F. & Kawohl, B. Overdetermined problems with possibly degenerate ellipticity, a geometric approach. Math. Z. 254, 117–132 (2006).

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