
Chapter 9
Second-Order
Elliptic Equations
with Two Space Variables
9.1 Laplace Equation ∆
2
w = 0
The Laplace equation is often encountered in heat and mass transfer theory, fluid mechan-
ics, elasticity, electrostatics, and other areas of mechanics and physics. For example, in
heat and mass transfer theory, this equation describes steady-state temperature 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 boundary value problem as