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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 $$S^ n$$, $$n\geq 3$$.

##### MSC:
 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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