Computational Methods and Function Theory 5 (2005), No. 2, 445--457
Copyright Heldermann Verlag 2005
Peter Ebenfelt
pebenfel@math.ucsd.edu , Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, U.S.A.
Michael Viscardi
mviscardi@earthlink.net , Josan Academy, 5524 Rabbit Ridge Rd., San Diego, CA 92130, U.S.A.
We study the Dirichlet problem for the Laplace operator in a simply connected bounded domain $\Omega$ in $\mathbb{R}^2 \cong \mathbb{C}$ with boundary data that are rational functions of one complex variable. Our main result is a characterization of those domains $\Omega$ for which all solutions are rational in terms of the Riemann mapping to the unit disk and in terms of the Bergman kernel of the domain.
Keywords: Dirichlet problem, Bergman kernel, Riemann mapping.
MSC 2000: 30E25, 35J25.
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