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An inverse Neumann problem. (English) Zbl 0623.35078
We prove that the only domains \(\Omega\) in \({\mathbb{R}}^ n\) (resp. \({\mathbb{N}}^ n)\) (n\(\geq 2)\) such that \(\partial \Omega\) is Lipschitz, connected and one of the following overdetermined boundary value problems (D) or (N) has infinitely many solutions. Are the balls in \({\mathbb{R}}^ n\) (resp. \({\mathbb{N}}^ n)\) \[ (D)\quad \Delta u+\alpha u=0\quad in\quad \Omega,\quad u=1,\quad \partial u/\partial n=0\quad on\quad \partial \Omega, \] \[ (N)\quad \Delta u+\alpha u=0\quad in\quad \Omega,\quad u=0,\quad \partial u/\partial n=1\quad on\quad \partial \Omega. \] These type of questions date back to Lord Rayleigh’s work in the 19th century. The problem (N) is related to the so-called Pompeiu’s problem in harmonic analysis.
We also observe in the paper that this theorem does not extend to subdomains of \(S^{2n+1}\). Recently counterexamples have been found for all \(S^ n\), \(n\geq 3\).

35R30 Inverse problems for PDEs
35N05 Overdetermined systems of PDEs with constant coefficients
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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