A characterization of the Khavinson-Shapiro conjecture via Fischer operators

TL;DR

This paper explores the Khavinson-Shapiro conjecture, demonstrating that the property of polynomial solutions to the Dirichlet problem in ellipsoids is equivalent to the surjectivity of a specific Fischer operator.

Contribution

It establishes a new characterization of the Khavinson-Shapiro conjecture through Fischer operators, linking geometric properties to operator surjectivity.

Findings

Property (KS) is equivalent to Fischer operator surjectivity for a domain.

Provides a new algebraic characterization of ellipsoids in relation to polynomial solutions.

Bridges geometric PDE properties with operator theory.

Abstract

The Khavinson-Shapiro conjecture states that ellipsoids are the only bounded domains in euclidean space satisfying the following property (KS): the solution of the Dirichlet problem for polynomial data is polynomial. In this paper we show that property (KS) for a domain $\Omega $ is equivalent to the surjectivity of a Fischer operator associated to the domain $\Omega .$