Thierry Gaspari, Mathématiques Pures de Bordeaux (MPB), UMR 5467 CNRS, Université Bordeaux 1, 351, cours de la Libération 33400 TALENCE, FRANCE E-mail address: gaspari@math.u-bordeaux.fr

Optimal Lipschitz Extensions

Let X be a Banach space, Ω be an open subset of X and u ∈ C(Ω, ℝ). We say that u is AML (Absolutely Minimizing Lipschitzian) if for all bounded open set V with \overline{V} ⊂ Ω, for all v ∈ C(\overline{V} , ℝ), we have

u = v on ∂V ⇒ Lip(u, V) ≤ Lip(v, V).

If u is AML in Ω and equals the function f on the boundary of Ω, then u is called an optimal Lipschitz extension of f in Ω. We study the relationship between the AML functions, the viscosity solutions of the infinity Laplacian partial differential equation, and the functions which satisfy another property called the comparison with cones. We prove that these notions are equivalent in any Hilbert space X. We also study the existence of optimal Lipschitz extensions.