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Existence and symmetry results for some overdetermined free boundary problems. (English) Zbl 1481.35289

Summary: In this paper, we prove that a domain which verifies some integral inequality is either (strictly) contained in the solution of some free boundary problem, or it coincides with an \(N\)-ball. We also present new overdetermined value problems which have an \(N\)-ball as a solution. To reach our results, we use an integral identity which involves the domain derivative of the solution of the corresponding Dirichlet problem.

MSC:

35N25 Overdetermined boundary value problems for PDEs and systems of PDEs
35A15 Variational methods applied to PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
35B50 Maximum principles in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35R35 Free boundary problems for PDEs
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