The Potential Equation

Abstract

The potential or Poisson equation

$$\Delta w (x) = f (x)$$

(6.1)

is perhaps the most studied equation in the partial differential equation literature. When the inhomogeneous term f(x) is identically zero, (6.1) is called the Laplace equation, or the harmonic equation, and the solution w is said to be a harmonic function. The potential equation is the simplest type of elliptic partial differential equations studied in this book, but because of the basic properties of the boundary integral operators involved, most of the theory and methodology developed here will actually be applicable to all second-order elliptic equations or systems. Therefore, the development in this chapter will also provide the essentialmathematical reasoning and procedures for treating all the other PDE in remaining chapters.