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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the determination of a measure by the orbits generated by its logarithmic potential


Authors: Dimitrios Betsakos and Simela Grigoriadou
Journal: Proc. Amer. Math. Soc. 134 (2006), 541-548
MSC (2000): Primary 31A15; Secondary 31A05, 70F99, 70K99
Published electronically: July 18, 2005
MathSciNet review: 2176023
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Abstract: We say that a logarithmic potential generates a curve in the plane if a unit mass traces the curve under the action of the potential. We consider the following problem: A one-parameter family of plane curves is given. We assume that these curves lie in the complement of a compact set $K$. Find all measures supported in $K$ whose potentials generate each of the given curves. We solve this problem when $K$ is the unit circle in three specific cases: (a) when the given curves are straight lines through the origin, (b) when the curves are straight lines through a point on the unit circle, and (c) when the curves are circles centered at the origin. The solution involves the Poisson integral and its boundary behavior.


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Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: betsakos@math.auth.gr

Simela Grigoriadou
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08000-7
PII: S 0002-9939(05)08000-7
Keywords: Logarithmic potential, inverse problem, reflection principle, harmonic function, Poisson integral.
Received by editor(s): May 28, 2004
Received by editor(s) in revised form: September 30, 2004
Published electronically: July 18, 2005
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2005 American Mathematical Society