A VISIT WITH THE -LAPLACE EQUATION MICHAEL G. CRANDALL1 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)| L|x - 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 right-hand 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 Collections: Mathematics