The Neumann Problem on Ellipsoids

TL;DR

This paper solves the Neumann boundary value problem for harmonic functions on ellipsoids in R^n with polynomial boundary data, providing existence conditions, polynomial solutions, and algorithms for computation.

Contribution

It establishes necessary and sufficient conditions for solutions, characterizes polynomial solutions, and develops algorithms for the Neumann problem on ellipsoids.

Findings

Solutions are polynomials of degree at most the boundary data polynomial.

A necessary and sufficient condition for solution existence is provided.

Algorithms for computing solutions to the Neumann and generalized Neumann problems are developed.

Abstract

The Neumann problem on an ellipsoid in R^n asks for a function harmonic inside the ellipsoid whose normal derivative is some specified function on the ellipsoid. We solve this problem when the specified function on the ellipsoid is a normalized polynomial (a polynomial divided by the norm of the normal vector arising from the definition of the ellipsoid). Specifically, we give a necessary and sufficient condition for a solution to exist, and we show that if a solution exists then it is a polynomial whose degree is at most the degree of the polynomial giving the specified function. Furthermore, we give an algorithm for computing this solution. We also solve the corresponding generalized Neumann problem and give an algorithm for computing its solution.