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Polynomial solutions to Dirichlet problems

Authors: Marc Chamberland and David Siegel
Journal: Proc. Amer. Math. Soc. 129 (2001), 211-217
MSC (2000): Primary 31A25
Published electronically: July 27, 2000
MathSciNet review: 1694451
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Abstract: The Dirichlet problem

\begin{eqnarraystar}\Delta u(x,y) & = & 0~~~\mbox{in}~ \mathbb{R}^2, u(x,y) & = & f(x,y)~~~\mbox{on}~\psi (x,y) = 0 \end{eqnarraystar}

is considered where the functions $f$ and $\psi$ are polynomials. The authors study the problem of determining which functions $\psi$ will admit polynomial solutions $u$ for any polynomial $f$. When one additionally requires the classical condition $\operatorname{deg} u \leq \operatorname{deg} f$, this forces $\operatorname{deg} \psi \leq 2$, and a complete classification is obtained. Some necessary conditions are obtained for the case $\operatorname{deg} \psi >2$.

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Additional Information

Marc Chamberland
Affiliation: Department of Mathematics and Computer Science, Grinnell College, Grinnell, Iowa 50112-0806

David Siegel
Affiliation: Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Received by editor(s): February 20, 1998
Received by editor(s) in revised form: April 5, 1999
Published electronically: July 27, 2000
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society

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