Sets with the Pompeiu property on the plane and on the sphere

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• 作者：V. V. Volchkov and Vit. V. Volchkov
• 关键词：Pompeiu problem ; Pompeiu property ; non ; Lipschitz boundary ; biangle ; spherical half ; disk ; Koch snowflake ; Morera ; type theorems ; Laplace operator
• 刊名：Mathematical Notes
• 出版年：2010
• 出版时间：February, 2010
• 年：2010
• 卷：87
• 期：1-2
• 页码：59-70
• 全文大小：607.4 KB

文摘

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.