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Computational Methods and Function Theory

, Volume 1, Issue 2, pp 387–401 | Cite as

An Inverse Problem for the Double Layer Potential

  • Peter Ebenfelt
  • Dmitry Khavinson
  • Harold S. Shapiro
Article

Abstract

We consider the problem of determining for which domains Ω ⊂ R n the number 1/2 is an eigenvalue for the operator taking a function on the boundary ∂Ω to the boundary values of its double layer potential. This question arises naturally in I. Fredholm’s solution to the Dirichlet problem for the Laplace operator in Ω. In two dimensions, the problem is equivalent to a matching problem for analytic functions which seems to be of independent interest. We show that the existence of a nontrivial solution for the matching problem characterizes the disk in a certain class of domains in the complex plane.

Keywords

Dirichlet problem double layer potential matching problem 

2000 MSC

31A25 31B20 

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Copyright information

© Heldermann Verlag 2001

Authors and Affiliations

  • Peter Ebenfelt
    • 1
  • Dmitry Khavinson
    • 2
  • Harold S. Shapiro
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaSan Diego, La JollaUSA
  2. 2.Department of MathematicsUniveristy of ArkansasFayettevilleUSA
  3. 3.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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