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An Inverse Problem for the Double Layer Potential

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093-0112, USA

    Peter Ebenfelt

  2. Department of Mathematics, Univeristy of Arkansas, Fayetteville, AR, 72701, USA

    Dmitry Khavinson

  3. Department of Mathematics, Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Harold S. Shapiro

Authors
  1. Peter Ebenfelt
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  2. Dmitry Khavinson
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  3. Harold S. Shapiro
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Corresponding author

Correspondence to Peter Ebenfelt.

Additional information

The first and second authors were partially supported by the NSF grants DMS-0100110 and DMS-9703915 respectively.

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Ebenfelt, P., Khavinson, D. & Shapiro, H.S. An Inverse Problem for the Double Layer Potential. Comput. Methods Funct. Theory 1, 387–401 (2001). https://doi.org/10.1007/BF03320998

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  • DOI: https://doi.org/10.1007/BF03320998

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