Existence of an Absolute Minimizer via Perron's Method

The existence of an absolute minimizer for a functional $$F(u,\Omega) = \underset{x \in \Omega}{ \text{ess sup}} \, f (x, u(x), Du(x))$$ is proved by using Perron's method. The function is assumed to be quasiconvex and uniformly coercive. This completes the result by T. Champion, L. De Pascale and F. Prinari [Gamma-convergence and absolute minimizers for supremal functionals, ESAIM Control Optim. Calc. Var. 10 (2004), No. 1, 14--27 (electronic)].