This article is cited in 1 paper

Sets with the Pompeiu Property on the Plane and on the Sphere

V. V. Volchkov, Vit. V. Volchkov

Donetsk National University

Abstract: We obtain new sufficient conditions under which a set on the plane has the Pompeiu property. This result allows us to construct first examples of domains with the Pompeiu property with non-Lipschitz (and even fractal) boundary. In addition, we study the problem of determining the least radius of the ball on the sphere in which a given set is a Pompeiu set. We obtain the solution of this problem in the case of a biangle and a spherical half-disk. We also consider some applications to questions of complex analysis.

Keywords: Pompeiu problem, Pompeiu property, non-Lipschitz boundary, biangle, spherical half-disk, Koch snowflake, Morera-type theorems, Laplace operator.

UDC: 517.444

Received: 16.06.2008

DOI: 10.4213/mzm5158

 English version: Mathematical Notes, 2010, 87:1, 59–70

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