 
Summary: A VISIT WITH THE LAPLACE EQUATION
MICHAEL G. CRANDALL1
Introduction
In these notes we present an outline of the theory of the archetypal L
variational
problem in the calculus of variations. Namely, given an open U IRn
and b C(U),
find u C(U) which agrees with the boundary function b on U and minimizes
(0.1) F(u, U):= Du L(U)
among all such functions. Here Du is the Euclidean length of the gradient Du of u. We
will also be interested in the "Lipschitz constant" functional as well. If K is any subset
of IRn
and u: K IR, its least Lipschitz constant is denoted by
(0.2) Lip(u, K):= inf {L IR : u(x)  u(y) Lx  y x, y K} .
Of course, inf = +. Likewise, if any definition such as (0.1) is applied to a function
for which it does not clearly make sense, then we take the righthand side to be +. One
has F(u, U) = Lip(u, U) if U is convex, but equality does not hold in general.
Example 2.1 and Exercise 2 below show that there may be many minimizers of F(·, U)
or Lip(·, U) in the class of functions agreeing with a given boundary function b on U.
While this sort of nonuniqueness can only take place if the functional involved is not
