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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)

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
Full Text: DOI
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