
Chapter 1 0
Second-Order
Elliptic Equations with
Three or More Space Variables
10.1 Laplace Equation ∆
3
w = 0
The three-dimensional Laplace equation is often encountered in heat and mass transfer
theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics.
For example, in heat and mass transfer theory, this equation describes stationary tempera-
ture distribution in the absence of heat sources and sinks in the domain under study.
A regular solution of the Laplace equation is called a harmonic function. The first
boundary value problem for the Laplace equation is often referred to as the D irichlet prob-
lem, and the second ...