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Another step toward the solution of the Pompeiu problem in the plane. (English) Zbl 0818.35136
Let $$\Omega \subset \mathbb{R}^ 2$$ be a bounded open set for which $$f \equiv 0$$ is the only continuous function on $$\mathbb{R}^ 2$$ such that (1) $$\int_{\sigma (\Omega)} f(x) dx = 0$$ for every rigid motion $$\sigma$$ of the plane. A bounded open set $$\Omega \subset \mathbb{R}^ 2$$ is said to have the Pompeiu property if there exist no nontrivial continuous function on $$\mathbb{R}^ 2$$ for which (1) holds. Several authors have determined bounded domains $$\Omega \subset \mathbb{R}^ 2$$ having the Pompeiu property.
The main result of this paper is the following: Let $$\Omega \subset \mathbb{R}^ 2$$ be a bounded simply-connected open set whose boundary is a closed simple curve parametrized by $$x(s) = (x_ 1(s), x_ 2(s))$$, $$s \in [- \pi, \pi]$$. Suppose that there exist $$M,N \in \mathbb{Z}$$ and $$a_ k \in \mathbb{C}$$, $$k = - M, \dots, N$$, with $$a_ M$$, $$a_ N \neq 0$$, such that $x_ 1 (s) + x_ 2 (s) = \sum^ N_{k= -M} a_ k e^{iks}.$ Let $$x_ 1 (z)$$, $$x_ 2 (z)$$ be the analytic extension of $$x_ 1 (s)$$ and $$x_ 2 (s)$$ satisfying $$(x_ 1'(z), x_ 2'(z))\neq (0,0) \in \mathbb{C}^ 2$$ for every $$z \in \mathbb{C}$$. Then $$\Omega$$ has the Pompeiu property.
Reviewer: G.Anger (Berlin)

MSC:
 35R30 Inverse problems for PDEs 31A25 Boundary value and inverse problems for harmonic functions in two dimensions 35P05 General topics in linear spectral theory for PDEs 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Keywords:
Pompeiu problem; Pompeiu property
Full Text:
References:
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