Minimal Lipschitz Extensions to Differentiable Functions Defined on a Hilbert Space

We generalize the Lipschitz constant to fields of affine jets and prove that such a field extends to a field of total domain \mathbbRⁿ{\mathbb{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 \mathbbRⁿ{\mathbb{R}^n} 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.