Arch. Rational Mech. Anal. 157 (2001) 255–283 Digital Object Identifier (DOI) 10.1007/s002050100133 The Euler Equation and Absolute Minimizers of L ∞ Functionals E.N.Barron,R.R.Jensen&C.Y.Wang Communicated by S. Müller Abstract The Aronsson-Euler equation for the functional F(u) = ess sup x∈ f(x,u(x),Du(x)), x ∈  ⊂ R n on W 1,∞ g (, R m ), i.e., W 1,∞ with boundary data g, is D x f(x,u(x),Du(x))f p (x,u(x),Du(x)) = 0. This equation has been derived for smooth absolute minimizers, i.e., a function which minimizes F on every subdomain. We prove in this paper that for m = 1,n ≧ 1, or n = 1,m ≧ 1 an absolute minimizer of F exists in W 1,∞ g (, R m ) and for m = 1,n ≧ 1 any absolute minimizer of F must be a viscosity solution of the Aronsson-Euler equation. 1. Introduction In [9] we introduced necessary and sufficient conditions for the L ∞ functional F(u; ) = ess sup x∈ f(x,u(x),Du(x)), u ∈ W 1,∞ (, R m ), to possess a minimizer in the class W 1,∞ (, R m ) with u achieving some given boundary data g ∈ W 1,∞ (, R m ) on the boundary of , a given bounded open subset of R n . These conditions all stemmed from the idea of a quasiconvex function. A function f : R n → R is quasiconvex if E γ =  x ∈ R n | f(x) ≦ γ  is convex for any γ ∈ R; equivalently, f(λx + (1 - λ)y) ≦ f(x) ∨ f(y), ∀ x,y ∈ R n ,λ ∈ (0, 1). Of course quasiconvexity is a generalization of convexity and the following theorem [8] is the analogue of Jensen’s inequality for quasiconvex functions.