Harmonic Polynomials and Dirichlet-Type Problems

We take a new approach to harmonic polynomials via differentiation. Surprisingly powerful results about harmonic functions can be obtained simply by differentiating the function |x|2−n and observing the patterns that emerge. This is one of our main themes and is the route we take to Theorem 1.7, which leads to a new proof of a harmonic decomposition theorem for homogeneous polynomials (Corollary 1.8) and a new proof of the identity in Corollary 1.10. We then discuss a fast algorithm for computing the Poisson integral of any polynomial. (Note: The algorithm involves differentiation, but no integration.) We show how this algorithm can be used for many other Dirichlet-type problems with polynomial data. Finally, we show how Lemma 1.4 leads to the identity in (3.2), yielding a new and simple proof that the Kelvin transform preserves harmonic functions. 1. Derivatives of |x|2−n Unless otherwise stated, we work in R, n > 2; the function |x|2−n is then harmonic and nonconstant on R \ {0}. (When n = 2 we need to replace |x|2−n with log |x|; the minor modifications needed in this case are discussed in Section 4.) Letting Dj denote the partial derivative with respect to the jth coordinate variable, we list here some standard differentiation formulas that will be useful later: Dj|x| = txj|x| ∆|x|t = t(t+ n− 2)|x|t−2 ∆(uv) = u∆v + 2∇u · ∇v + v∆u. The first two formulas are valid on R \ {0} for every real t, while the last formula holds on any open set where u and v are twice continuously differentiable (and real valued); as usual, ∆ denotes the Laplacian and ∇ denotes the gradient. 1991 Mathematics Subject Classification. 31B05. The first author was partially supported by the National Science Foundation.