Beginning Partial Differential Equations, 3rd Edition

Chapter 4

Dirichlet and Neumann Problems

4.1 Laplace’s Equation and Harmonic Functions

In two dimensions, the heat equation is

In the limit as t → ∞, and in the absence of sources or sinks of energy, the effects of the initial conditions are no longer felt in the solution and we think of u_t as zero to obtain Laplace’s equation

Laplace’s equation is also known as the potential equation and also as the steady state heat equation.

In three space variables, Laplace’s equation is

In vector calculus, the del symbol denotes the gradient operator, where

and, in three dimensions,

Now Laplace’s equation can be written

² is read del squared. The del symbol is also called nabla after the Greek word ...