Principles of Comparison with Distance Functions for Absolute Minimizers

We extend the principle of comparison with cones introduced by M. G. Crandall, L. C. Evans and R. F. Gariepy [Calc. Var. Partial Diff. Equations 13 (2001) 123--139] for the Minimizing Lipschitz Extension Problem to a wide class of supremal functionals. This gives a geometrical characterization of the absolute minimizers (optimal solutions whose minimality is local). Some application to the stability of absolute minimizers with respect to the Γ-convergence is given. A variation of the basic idea also allows to characterize the minimal Lipschitz extensions in length metric spaces.