The $R^N$ Dirichlet Problem

Abstract

In this chapter the notation is made simpler by considering harmonic functions which are complex valued functions and have real and imaginary parts satisfying the Laplace equation in $N$-variables. The Dirichlet problem is to find such a function of $N$ variables, satisfying the Laplace equation in the interior of a domain $D$ and being equal to a given continuous complex-valued function $g$ on the boundary $\Gamma$ of $D$. The case of interest in applications is $N$=3, but there is no essential difference in considering general $N$. In addition to specifying that the bounded open set $D$ not have any holes, bubbles in three dimensions, which was enough in two dimensions, additional conditions are necessary in three or higher dimensions, and the relevant Poincaré condition is discussed. The proof uses the Hahn-Banach Theorem applied in the space $C(K)$ of complex-valued continuous functions defined on the boundary K of the domain. In the final result it is shown that there is an approximate solution using any one chosen function $f(z)$, which is not a polynomial and which has a power series convergent (about z = 0). This approximate solution is a finite sum of the form ${c_n}f({a^{(n)}} \cdot x + {b^n} \cdot x + {d_n})$ where ${c_n}$ and ${d_n}$ are complex numbers, $a^{(n)}$ and $b^n$ are vectors in $R^N$ with the same length, ||${a^{(n)}}$|| = ||${b^{(n)}}$||, which are perpendicular $a^{(n)}$ $\cdot$ $b^n$ = 0, and x is a point in $R^N$. The proof relies on a lemma which shows how to construct a harmonic function in $N$ variables from a harmonic function in two variables, which will be applied to the real and imaginary parts of $f(z)$, and a representation theorem for the space of harmonic polynomials in $N$-variables which are homogeneous of degree $m$. Note that this result demonstrates the ability of a function of two variables, $f(z)$, to generate an approximate solution of the Dirichlet problem in $N$-variables.