Polynomial solutions to Dirichlet problems

The Dirichlet problem

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 $\operatorname{deg} \psi >2$.