Extremal Extensions for $m$-jets from $\R$ to $\R^n$

Erwan Le Gruyer (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We characterize the Lipschitz constant for the $m$-fields ($m \in \N$) from $\R$ to $\R^n$. This work completes the results of J. Favard \cite{Favard} and G. Glaeser \cite{Glaeser1} (see also \cite{Legruyer1}).\\ Let us consider a $m$-field $U$. Our problem is to solve $$ \inf \{ \mbox{Lip}(g^{(m)}) \mbox{ : } g \mbox{ is an } m\mbox{-Lipschitz extension of }U \} , $$ where $\mbox{Lip}$ is the Lipschitz constant, and to characterize the \textit{extremal} extension $f$, according to Favard's terminology \cite{Favard}, for which the above infimum is attained. The expression of the extremal extension contains the antiderivative of a rational function where the numerator is a polynomial and the denominator is the Euclidean norm of this polynomial. We further study the stability of this solution.

Mots clés: Minimal; Lipschitz Extension; Differentiable Function

Domaines: Analyse numérique [math.NA]

Erwan Le Gruyer. Extremal Extensions for $m$-jets from $\R$ to $\R^n$. 2010. ⟨hal-00489064⟩