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


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.


Dirichlet problem double layer potential matching problem 

2000 MSC

31A25 31B20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [C]
    V. G. Cherednichenko, Inverse Logarithmic Potential Problem, VSP Press, Utrecht, 1996.MATHGoogle Scholar
  2. [Da]
    P. J. Davis, The Schwarz Function And Its Applications, The Carus Mathematical Monographs, No. 17. The Mathematical Association of America, Buffalo, N.Y., 1974.MATHGoogle Scholar
  3. [Du]
    P. Duren, Theory of Hp-Spaces, Pure and Applied Math. 39, Acad. Press, New York, 1970.Google Scholar
  4. [E]
    P. Ebenfelt, Singularities encountered by the analytic continuation of solutions to Dirichlet’s problem, Complex Variables 20 (1992), 75–92.MathSciNetCrossRefMATHGoogle Scholar
  5. [IK]
    T. Iwaniec and R. Kosecki, Sharp estimates for complex potentials and quasiconformal mappings, preprint, 1989.Google Scholar
  6. [K]
    O. D. Kellogg, Foundations of Potential Theory, reprinted from 1929, ed. by Dover Publ., 1953.Google Scholar
  7. [KS]
    N. Kerzman and E. M. Stein, The Cauchy kernel, the Szegö kernel, and the Riemann mapping function, Math. Ann 236 (1978), 85–93.MathSciNetCrossRefMATHGoogle Scholar
  8. [Ma]
    A. I. Markushevich, Theory of Functions of a Complex Variable, Vol. I, II, III., (Translated and edited by Richard A. Silverman) Second English edition, Chelsea Publishing Co., New York, 1977.Google Scholar
  9. [Mu]
    N. I. Muskhelishvili, Singular Integral Equations (in Russian), Third Ed., Nauka, Moscow, 1968.MATHGoogle Scholar
  10. [S]
    H. S. Shapiro, The Schwarz Function and its Generalization to Higher Dimensions, Wiley-Interscience, 1992.Google Scholar

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

Personalised recommendations