Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

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 $K$. 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.


References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 31A15, 31A05, 70F99, 70K99

Retrieve articles in all journals with MSC (2000): 31A15, 31A05, 70F99, 70K99


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