Computational Methods and Function Theory 1 (2001), No. 2, 387--401
Copyright Heldermann Verlag 2001
Peter Ebenfelt
pebenfel@math.ucsd.edu, Department of Mathematics, University of California, San Diego, La Jolla, CA 92093--0112, U.S.A.
Dmitry Khavinson
dmitry@comp.uark.edu, Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, U.S.A.
Harold S. Shapiro
shapiro@math.kth.se, Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden.
We consider the problem of determining for which domains $\Omega\subset \R^n$ the number 1/2 is an eigenvalue for the operator taking a function on the boundary $\partial \Omega$ 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 $\Omega$. 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.
MSC 2000: 31A25, 31B20.
[FullText-pdf (224 K)] [FullText-ps (284 K)]