A Visit with the ∞-laplace Equation

(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 IR 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 strictly convex, it is more significant here that the functionals are “not local.” Let us explain what we mean in contrast with the Dirichlet functional