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A new result on the Pompeiu problem

Authors:	R Dalmasso

Institution:	Laboratoire LMC-IMAG, Equipe EDP, BP 53, F-38041 Grenoble Cedex 9, France

Abstract:	A nonempty bounded open set $\Omega \subset {\mathbb{R}}^{n}$ ( $n \geq 2$ ) is said to have the Pompeiu property if and only if the only continuous function on ${\mathbb{R}}^{n}$ for which the integral of over $\sigma (\Omega )$ is zero for all rigid motions $\sigma$ of ${\mathbb{R}}^{n}$ is $f \equiv 0$ . We consider a nonempty bounded open set $\Omega \subset {\mathbb{R}}^{n}$ $(n \geq 2)$ with Lipschitz boundary and we assume that the complement of $\overline{\Omega }$ is connected. We show that the failure of the Pompeiu property for $\Omega$ implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in ${\mathbb{R}}^{n}$ , $n \geq 3$ , has the Pompeiu property. So far the result was proved only for solid tori in ${\mathbb{R}}^{4}$ . We also examine the case of planar domains. Finally we extend the example of solid tori to domains in ${\mathbb{R}}^{n}$ bounded by hypersurfaces of revolution.

Keywords:	Pompeiu problem Schiffer's conjecture

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