Quasiminima of the Lipschitz extension problem

In this paper, we extend the notion of quasiminimum to the framework of supremum functionals by studying the model case which governs the real analysis problem of finding optimal Lipschitz extensions. Using a characterization involving the concept of comparison with cones, we obtain a Harnack inequality, Lipschitz estimates and various convergence and stability properties for the quasiminima. Several examples of quasiminima are also given. Mathematics Subject Classification (2000) 47J20, 49N60, 35B65