On Absolutely Minimizing Lipschitz Extensions and PDE ${\mathrm{\Delta }}_{i}nfty=0$

We prove the existence of Absolutely Minimizing Lipschitz Extensions by a method which differs from those used by G. Aronsson in general metrically convex compact metric spaces and R. Jensen in Euclidean spaces. Assuming Jensen’s hypotheses, our method yields numerical schemes for computing, in euclidean $\mathbb{R}$, the solution of viscosity of equation ${\mathrm{\Delta }}_{\mathrm{\infty }}=0$ with Dirichlet’s condition.