A tour of the theory of absolutely minimizing functions

Abstract

Abstract. These notes are intended to be a rather complete and self-contained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding existence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely self-contained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial differential equations are illustrated here in an especially clean manner, free from burdensome technicalities -indeed, usually free from partial differential equations themselves. These include a priori continuity estimates, the Harnack inequality, Perron's method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unified summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some significant analytical tools.

Everywhere differentiability of infinity harmonic functions

“…Likewise for z ε := 1 2 (u ε ) 2 we have the identity 6) according to (2.1). Select a smooth function ζ such that ζ ≡ 1 on V , and ζ ≡ 0 near ∂U , and put…”

“…This highly degenerate nonlinear PDE arises as a variational equation in the "calculus of variations in the sup-norm" (Crandall [3], Aronsson et al [2]) and also appears in stochastic "tug-of-war" game theory (Peres et al [10]). A viscosity solution u is called an infinity harmonic function.…”

An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions

“…The next lemma allows us to modify solutions of the PDE (1) to obtain solutions of the finite difference inequalities (2). We use the notation u, x ∈ ε , which allows us to write εS…”

A Formal Derivation of the Aronsson Equations for Symmetrized Gradients

“…However, in this case the left hand side of (8) also vanishes on this set. Formally letting p → ∞ in (8) now leads to the infinity-Laplace equation (6). This computation can be made rigorous (see, e.g., Bhattacharya, DiBenedetto, and Manfredi [8]): if u p is a (weak) solution of (4), then u p converges uniformly on compact subsets…”

“…We recall that a continuous function u is a viscosity solution of (6) if on one hand, for every local maximum point x ∈ Ω of u − ϕ, where ϕ is a C 2 function in a neighborhood of x and u(x) = ϕ(x), we have −∆ ∞ ϕ(x) ≤ 0, and on the other hand, whenever x ∈ Ω is a local minimum point of u − ϕ, where ϕ is a C 2 function in a neighborhood of x and u(x) = ϕ(x), we have −∆ ∞ ϕ(x) ≥ 0. An excellent account on the theory of infinity harmonic functions can be found in Aronsson, Crandall, and Juutinen [6].…”