On the Dirichlet problem in the plane with polynomial data

TL;DR

This paper proves that certain polynomial and rational Dirichlet problems in the plane uniquely characterize ellipses and discs, extending classical results through purely algebraic methods.

Contribution

It establishes algebraic proofs that connect polynomial and rational solutions of Dirichlet problems to geometric shapes like ellipses and discs.

Findings

If polynomial boundary data solutions exist, the domain is an ellipse.

Existence of rational solutions implies the domain is a disc.

Generalizes previous results by Bell, Ebenfelt, Khavinson, and Shapiro.

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.