Local Overdetermined Linear Elliptic Problems in Lipschitz Domains with Solutions Changing Sign
Bruno Canuto, Diego F. Rial · OpenstarTs (Univeristy of Trieste https://www.units.it/) · 2009
We prove that the only domain Ω such that there exists a solution to the following overdetermined problem ∆u+ ω 2 u = -1 in Ω, u = 0 on ∂Ω, and ∂ n u = c on ∂Ω, is the ball B 1 , independently on the sign of u, if we assume that the boundary ∂Ω is a perturbation (no necessarily regular) of the unit sphere ∂B 1 of R n .Here ω 2 = (λ n ) n≥1 (the eigenvalues of -∆ in B 1 with Dirichlet boundary conditions), and ω / ∈ Λ, where Λ is a enumerable set of R + , whose limit points are the values λ 1m , for some integer m ≥ 1, λ 1m being the m th -zero of the first-order Bessel function I 1 .