Tensor rectifiable G-flat chains

Goldman, Michael; Merlet, Benoît

Type de document :

Pré-publication ou Document de travail: Autre communication scientifique (congrès sans actes - poster - séminaire...)

Titre :

Tensor rectifiable G-flat chains

Auteur(s) :

Goldman, Michael [Auteur]
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Merlet, Benoît [Auteur]

Reliable numerical approximations of dissipative systems [RAPSODI]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Discipline(s) HAL :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

Résumé en anglais : [en]

A rigidity result for normal rectifiable $k$-chains in $\mathbb{R}^n$ with coefficients in an Abelian normed group is established. Given some decompositions $k=k_1+k_2$, $n=n_1+n_2$ and some rectifiable $k$-chain $A$ in ...
Lire la suite >A rigidity result for normal rectifiable $k$-chains in $\mathbb{R}^n$ with coefficients in an Abelian normed group is established. Given some decompositions $k=k_1+k_2$, $n=n_1+n_2$ and some rectifiable $k$-chain $A$ in $\mathbb{R}^n$, we consider the properties:(1) The tangent planes to $\mu_A$ split as $T_x\mu_A=L^1(x)\times L^2(x)$ for some $k_1$-plane $L^1(x)\subset\mathbb{R}^{n_1}$ and some $k_2$-plane $L^2(x)\subset\mathbb{R}^{n_2}$.(2) $A=A_{\vert\Sigma^1\times\Sigma^2}$ for some sets $\Sigma^1\subset\mathbb{R}^{n_1}$, $\Sigma^2\subset\mathbb{R}^{n_2}$ such that $\Sigma^1$ is $k_1$-rectifiable and $\Sigma^2$ is $k_2$-rectifiable (we say that $A$ is $(k_1,k_2)$-rectifiable).The main result is that for normal chains, (1) implies (2), the converse is immediate. In the proof we introduce the new groups of tensor flat chains (or $(k_1,k_2)$-chains) in $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ which generalize Fleming's $G$-flat chains. The other main tool is White's rectifiable slices theorem. We show that on the one hand any normal rectifiable chain satisfying~(1) identifies with a normal rectifiable $(k_1,k_2)$-chain and that on the other hand any normal rectifiable $(k_1,k_2)$-chain is $(k_1,k_2)$-rectifiable.Lire moins >

Langue :

Anglais

Projet ANR :

Optimisation de forme
Centre Européen pour les Mathématiques, la Physique et leurs Interactions

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