Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Polynomial solutions to Dirichlet problems

Author(s): Marc Chamberland; David Siegel
Journal: Proc. Amer. Math. Soc. 129 (2001), 211-217.
MSC (2000): Primary 31A25
Posted: July 27, 2000
MathSciNet review: 1694451
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Abstract | References | Similar articles | Additional information

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

and $\psi$ are polynomials. The authors study the problem of determining which functions $\psi$ will admit polynomial solutions

for any polynomial

. 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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