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Perturbation of domains in the Pompeiu problem. (English) Zbl 0847.35090
Summary: An old problem in integral geometry called the Pompeiu problem is closely related to the existence of a solution of the overdetermined Neumann problem: \[ \Delta+ \lambda u= 0\quad \text{in } \Omega,\quad {\partial u\over \partial \nu}= 0,\;u\equiv \text{constant}\quad \text{on } \partial\Omega.(N)_\lambda \] It is easy to see that \((N)_\lambda\) has a nontrivial solution if \(\Omega\) is a ball. In this paper, we shall give a quantitative estimate of the following statement in terms of a one parameter family of domains and some special values of Bessel functions: If \(\Omega\) is sufficiently ‘close to’ a ball and if \((N)_\lambda\) has a nontrivial solution, which is not too large, then \(\Omega\) must be a ball.

35N05 Overdetermined systems of PDEs with constant coefficients
35J25 Boundary value problems for second-order elliptic equations
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