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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

New results on the Pompeiu problem


Authors: Nicola Garofalo and Fausto Segàla
Journal: Trans. Amer. Math. Soc. 325 (1991), 273-286
MSC: Primary 35R30; Secondary 31B20, 35J05
MathSciNet review: 994165
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Abstract: Let $ {p_N}(w) = \sum\nolimits_{k = 0}^N {{a_k}{w^k}} $, $ w \in \mathbb{C}$, $ N \in \mathbb{N}$, be a polynomial with complex coefficients. In this paper we prove that if $ D \subset {\mathbb{R}^2}$ is a simply-connected bounded open set whose boundary is a closed, simple curve parametrized by $ x(s) = {x_1}(s) + i{x_2}(s) = {p_N}({e^{is}})$, $ s \in [ - \pi ,\pi ]$, then $ D$ has the Pompeiu property unless $ N = 1$ and $ {p_1}(w) = {a_1}w + {a_2}$ in which case $ D$ is a disk. This result supports the conjecture that modulo sets of zero two-dimensional Lebesgue measure, the disk is the only simply-connected, bounded open set which fails to have the Pompeiu property.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-0994165-X
PII: S 0002-9947(1991)0994165-X
Article copyright: © Copyright 1991 American Mathematical Society