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 | 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.

**1.**G. Bozis,*Szebehely’s inverse problem for finite symmetrical material concentrations*, Astronom. and Astrophys.**134**(1984), no. 2, 360–364. MR**748011****2.**George Bozis,*The inverse problem of dynamics: basic facts*, Inverse Problems**11**(1995), no. 4, 687–708. MR**1345999****3.**Gerald B. Folland,*Real analysis*, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR**1681462****4.**O.D. Kellogg,*Foundations of Potential Theory*. Dover Publications 1953.**5.**Murray H. Protter and Hans F. Weinberger,*Maximum principles in differential equations*, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR**762825****6.**Thomas Ransford,*Potential theory in the complex plane*, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR**1334766****7.**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157**

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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