Polynomial solutions to Dirichlet problems

The Dirichlet problem Delta u(x, y) = 0 in R-2, u(x, y) = f(x, y) on psi(x, y) = 0 is considered where the functions f and psi are polynomials. The authors st udy the problem of determining which functions will admit polynomial soluti ons u for any polynomial f. When one additionally requires the classical co ndition deg u less than or equal to deg f, this forces deg psi less than or equal to 2, and a complete classification is obtained. Some necessary cond itions are obtained for the case deg psi > 2.