Sup–Inf explicit formulas for minimal Lipschitz extensions for 1-fields on RnRn

We study the relationship between the Lipschitz constant of 1-field introduced in [12] and the Lipschitz constant of the gradient canonically associated with this 1-field. Moreover, we produce two explicit formulas which are two extremal minimal Lipschitz extensions for 1-fields. As a consequence of the previous results, for the problem of minimal extension by Lipschitz continuous functions from RmRm to RnRn, we produce explicit formulas similar to those of Bauschke and Wang (see [7]). Finally, we show that Wells's extensions (see [24]) of 1-fields are absolutely minimal Lipschitz extensions when the domain of 1-field to expand is finite. We provide a counter-example showing that this result is false in general.