Local overdetermined linear elliptic problems in Lipschitz domains with solutions changing sign
We prove that the only domain Ω such that there exists a solution to the following overdetermined problem Δu+ω2u=-1 in Ω, u=0 on ∂Ω, and ∂nu=c on ∂Ω, is the ball B1, independently on the sign of u, if we assume that the boundary ∂Ω is a perturbation (no necessarily regular) of the unit sphere ∂B1 of...
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2008
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00494704_v40_n_p1_Canuto http://hdl.handle.net/20.500.12110/paper_00494704_v40_n_p1_Canuto |
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| Sumario: | We prove that the only domain Ω such that there exists a solution to the following overdetermined problem Δu+ω2u=-1 in Ω, u=0 on ∂Ω, and ∂nu=c on ∂Ω, is the ball B1, independently on the sign of u, if we assume that the boundary ∂Ω is a perturbation (no necessarily regular) of the unit sphere ∂B1 of ℝn. Here ω2 ≠ (λn)n≥1 (the eigenvalues of -Δ in B1 with Dirichlet boundary conditions), and ω ∉ Λ, where Λ is a enumerable set of ℝ+, whose limit points are the values λ1m, for some integer m ≤ 1, λ1m being the mth-zero of the first-order Bessel function I1. © 2009, EUT Edizioni Universita di Trieste. |
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