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), 541548
MSC (2000):
Primary 31A15; Secondary 31A05, 70F99, 70K99
Published electronically:
July 18, 2005
MathSciNet review:
2176023
Fulltext PDF Free Access
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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 oneparameter family of plane curves is given. We assume that these curves lie in the complement of a compact set . Find all measures supported in whose potentials generate each of the given curves. We solve this problem when 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/S0002993905080007
PII:
S 00029939(05)080007
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
