The Euler Equation and¶Absolute Minimizers of L∞ Functionals

Abstract:

The Aronsson-Euler equation for the functional

on W _g ^{1, ∞}(Ω, ℝ^m, i.e., W ^{1, ∞} with boundary data g, is

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 _g ^{1, ∞}(Ω, ℝ^m and for m= 1, n≧ 1 any absolute minimizer of F must be a viscosity solution of the Aronsson-Euler equation.