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On the Solution of the Dirichlet Problem with Rational Holomorphic Boundary Data

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Abstract

We study the Dirichlet problem for the Laplace operator in a simply connected bounded domain Ω in ℝ = C with boundary data that are rational functions of one complex variable. Our main result is a characterization of those domains Ω 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.

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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. Josan Academy, 5524 Rabbit Ridge Rd., San Diego, CA, 92130, USA

    Michael Viscardi

Authors
  1. Peter Ebenfelt
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  2. Michael Viscardi
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Correspondence to Peter Ebenfelt.

Additional information

The first author is supported in part by DMS-0401215.

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Ebenfelt, P., Viscardi, M. On the Solution of the Dirichlet Problem with Rational Holomorphic Boundary Data. Comput. Methods Funct. Theory 5, 445–457 (2006). https://doi.org/10.1007/BF03321109

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  • DOI: https://doi.org/10.1007/BF03321109

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