Extremal Extensions for mm-jets from R\R to Rn\R^n

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

Kirszbraun extension on connected finite graph

We prove that the tight function introduced Sheffield and Smart (2012) is a Kirszbraun extension. In the real-valued case we prove that Kirszbraun extension is unique. Moreover, we produce a simple algorithm which calculates efficiently the value of Kirszbraun extension in polynomial time

Minimal Lipschitz Extensions to differentiable functions

We generalize the Lipschitz constant to Whitney's functions and prove that any Whitney's function defined on a non-empty subset of $\bold R^n$ extends to a Whitney's function of domain $\bold R^n$ with the same constant. The proof uses an argument which refines the one used by Kirszbraun in the continuous case and, for this reason, holds only for $\bold R^n$ equiped with the euclidean norm. This constant is exactly the Lipschitz constant of the gradient of the extension and, therefore, this extension is minimal. We continue the paper with a first approach of the absolutely minimal Lipschitz extension problem in the differentiable case

A general theorem of existence of quasi absolutely minimal Lipschitz extensions

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier "quasi" to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.Comment: 33 pages. v3: Correction to Example 2.4.3. Specifically, alpha-H\"older continuous functions, for alpha strictly less than one, do not satisfy (P3). Thus one cannot conclude that quasi-AMLEs exist in this case. Please note that the error remains in the published version of the paper in Mathematische Annalen. v2: Several minor corrections and edits, a new appendix (Appendix A

Explicit formulas for C1,1C^{1,1} Glaeser-Whitney extensions of 1-fields in Hilbert spaces

We give a simple alternative proof for the $C^{1,1}$--convex extension problem which has been introduced and studied by D. Azagra and C. Mudarra [2]. As an application, we obtain an easy constructive proof for the Glaeser-Whitney problem of $C^{1,1}$ extensions on a Hilbert space. In both cases we provide explicit formulae for the extensions. For the Gleaser-Whitney problem the obtained extension is almost minimal, that is, minimal up to a factor $\frac{1+\sqrt{3}}{2}$ in the sense of Le Gruyer [15]

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

Extremal extension for m-jets of one variable with range in a Hilbert space

International audienceWe generalize to Hilbert spaces a theorem of Glaeser concerning minimal Lipschitz extensions of m-jets of one variable with range in R. The results contained in this paper can be seen as a small contribution to the general problem of the minimal Lipschitz extensions from m-jets for a Hilbert space to another Hilbert space

Minimal Lipschitz extensions to differentiable functions defined on a Hilbert space

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

On Absolutely Minimizing Lipschitz Extensions and PDE Δinfty=0\Delta_infty = 0

2004-02International audienceWe prove the existence of Absolutely Minimizing Lipschitz Extensions by a method which differs from those used by G. Aronsson in general metrically convex compact metric spaces and R. Jensen in Euclidean spaces. Assuming Jensen's hypotheses, our method yields numerical schemes for computing, in euclidean $\mathbb R$, the solution of viscosity of equation $\Delta_\infty=0$ with Dirichlet's condition