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

DOI:
https://doi.org/10.1090/S0002-9939-05-08000-7

Published electronically:
July 18, 2005

MathSciNet review:
2176023

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Abstract | References | Similar Articles | Additional Information

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 **. 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:
https://doi.org/10.1090/S0002-9939-05-08000-7

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