 
Summary: THE DIRICHLET PROBLEM ON QUADRATIC SURFACES
SHELDON AXLER, PAMELA GORKIN, AND KARL VOSS
Abstract. We give a fast, exact algorithm for solving Dirichlet problems with
polynomial boundary functions on quadratic surfaces in Rn such as ellipsoids,
elliptic cylinders, and paraboloids. To produce this algorithm, first we show
that every polynomial in Rn can be uniquely written as the sum of a harmonic
function and a polynomial multiple of a quadratic function, thus extending
a theorem of Ernst Fischer. We then use this decomposition to reduce the
Dirichlet problem to a manageable system of linear equations. The algorithm
requires differentiation of the boundary function, but no integration. We also
show that the polynomial solution produced by our algorithm is the unique
polynomial solution, even on unbounded domains such as elliptic cylinders and
paraboloids.
1. Introduction
In this paper we present a fast, exact algorithm for solving Dirichlet problems
with polynomial boundary functions on a quadratic surface in Rn
(n 2). To
illustrate the kind of Dirichlet problem we study, fix b = (b1, . . . , bn) Rn
. For
x = (x1, . . . , xn) Rn
