MINIMAL LIPSCHITZ EXTENSIONS TO DIFFERENTIABLE FUNCTIONS DEFINED ON A HILBERT SPACE.

We generalize the Lipschitz constant to fields of a. ne jets and prove that such a field extends to a field of total domain R(n) with the same constant. This result may be seen as the analog for fields of the minimal Kirszbraun's extension theorem for Lipschitz functions and, therefore, establishes a link between Kirszbraun's theorem and Whitney's theorem. In fact this result holds not only in Euclidean Rn but also in general (separable or not) Hilbert space. We apply the result to the functional minimal Lipschitz differentiable extension problem in Euclidean spaces and we show that no Brudnyi-Shvartsman-type theorem holds for this last problem. We conclude with a first approach of the absolutely minimal Lipschitz extension problem in the differentiable case which was originally studied by Aronsson in the continuous case.