The Khavinson-Shapiro conjecture for domains with a boundary consisting of algebraic hypersurfaces

TL;DR

This paper investigates the Khavinson-Shapiro conjecture, demonstrating that certain algebraic boundary configurations prevent a domain from satisfying the property that polynomial boundary data yields polynomial solutions.

Contribution

It proves that domains with boundaries containing at least three algebraic hypersurfaces sharing a common point do not satisfy the Khavinson-Shapiro property.

Findings

Domains with such boundary configurations lack property (KS)

The boundary structure influences the polynomial solvability of the Dirichlet problem

Provides conditions under which the conjecture does not hold

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 a domain does not have property (KS) provided the boundary contains at least three differrent irreducible algebraic hypersurfaces for which two of them have a common point.