On the Dirichlet problem in the plane with polynomial data

Abstract

Let $\Omega\subset\mathbb{C}$ be a bounded domain such that there exists an algebraic harmonic function of degree two vanishing on the boundary of $\Omega.$ Then we show that the Khavinson-Shapiro conjecture holds for $\Omega:$ if the Dirichlet problem on $\Omega$ with all polynomial boundary data have polynomial solutions, then $\Omega$ must be an ellipse. We also prove that if there exists a rational function with a singularity in $\Omega$, such that the Dirichlet problem for its restriction on $\partial\Omega$ along with all polynomial functions have rational solutions, then $\Omega$ must be a disc. This generalizes a well-known result by Bell, Ebenfelt, Khavinson, and Shapiro. Our proofs are purely algebraic.