Poisson’s Equation and the Newtonian Potential

Abstract

In Chapter 2 we introduced the fundamental solution Г of Laplace’s equation given by

$$\Gamma (x - y) = \Gamma \left( {\left| {x - y} \right|} \right) = \left\{ \begin{gathered} \frac{1}{{n\left( {2 - n} \right){{\omega }_{n}}}}{{\left| {x - y} \right|}^{{2 - n}}}, n > 2 \hfill \\ \frac{1}{{2\pi }}\log \left| {x - y} \right|, n = 2. \hfill \\ \end{gathered} \right.$$

(4.1)

. For an integrable function f on a domain Ω, the Newtonian potential of f is the function w defined on ℝⁿ by

$$ w(x) = \int\limits_{\Omega } \Gamma (x - y)f(y)dy $$

(4.2)

. From Green’s representation formula (2.16), we see that when ∂Ω is sufficiently smooth a C ²(ΩΩ) function may be expressed as the sum of a harmonic function and the Newtonian potential of its Laplacian. It is not surprising therefore that the study of Poisson’s equation Δu = f can largely be effected through the study of the Newtonian potential of f. This chapter is primarily devoted to the estimation of derivatives of the Newtonian potential. As well as leading to existence theorems for the classical Dirichlet problem for Poisson’s equation, these estimates form the basis for the Schauder or potential theoretic approach to linear elliptic equations treated in Chapter 6.