Local Overdetermined Linear Elliptic Problems in Lipschitz Domains with Solutions Changing Sign

BRUNO CANUTO; DIEGO RIAL

RENDICONTI DELL'ISTITUTO DI MATEMATICA DELL'UNIVERSITÀ DI TRIESTE

UNIVERSITÁ DI TRIESTE

Lugar: Trieste; Año: 2009 vol. 40 p. 1 - 27

0049-4704

Abstract. We prove that the only domain Ω such that there exists a solution to the following overdetermined problem ∆u+ 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 −∆ 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.