The Euler equation and absolute minimizers of L-infinity functionals

The Aronsson-Euler equation for the functional F(u) = ess(x is an element of Omega) sup f (x, u(x), Du(x)), x is an elemen t of Omega subset of R-n on W-g(1,infinity) (Omega ,R-m), i.e., W-1,W-infinity 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 func tion which minimizes F on every subdomain. We prove in this paper that for m = 1, n greater than or equal to 1, or n = 1, m greater than or equal to 1 an absolute minimizer of F exists in W-g(1,infinity)(Omega ,R-m) and for m = 1, n greater than or equal to 1 any absolute minimizer of F must be a vi scosity solution of the Aronsson-Euler equation.