Some results of the Lipschitz constant of 1-Field on Rn\mathbb{R}^n

We study the relations between the Lipschitz constant of $1$-field and the Lipschitz constant of the gradient canonically associated with this $1$-field. Moreover, we produce two explicit formulas that make up Minimal Lipschitz extensions for $1$-field. As consequence of the previous results, for the problem of minimal extension by continuous functions from $\mathbb{R}^m$ to $\mathbb{R}^n$, we also produce analogous explicit formulas to those of Bauschke and Wang. Finally, we show that Wells's extensions of $1$-field are absolutely minimal Lipschitz extension when the domain of $1$-field to expand is finite. We provide a counter-example showing that this result is false in general.Comment: E.L.G. and T.V.P. are partially supported by the ANR (Agence Nationale de la Recherche) through HJnet projet ANR-12-BS01-0008-0