Solutions of Homogeneous Elliptic Equations

Extract

We consider an elliptic partial differential equation with constant coefficients and zero on the right hand side. It is well known [1] that every solution of such an equation can be approximated uniformly on each compact set by a sum of products of polynomials and exponential functions which satisfy the equation. Furthermore, if one assumes that the polynomial operator is homogeneous, then the approximation can be made with polynomials alone. It is our purpose to show, in the latter case, when the number of variables is two, that each solution can be written as an infinite series in certain specific polynomials. Our method is to factor the polynomial and build up the solution in terms of solutions of first degree equations.