## 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.

### MSC:

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

### Keywords:

Pompeiu problem; overdetermined Neumann problem
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