Polynomial solutions to Dirichlet problems

Abstract:

The Dirichlet problem \begin{align*} \Delta u(x,y) &= 0 && \text {in $\mathbb {R}^2$},\\ u(x,y) &= f(x,y) && \text {on $\psi (x,y) = 0$} \end{align*} 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 $\operatorname {deg} \psi >2$.