Counterexamples in calculus of variations in <i>L</i><sup>∞</sup> through the vectorial Eikonal equation

Katzourakis, Nikos; Shaw, Giles

doi:10.1016/j.crma.2018.04.010

Partial differential equations/Calculus of variations

Counterexamples in calculus of variations in L^∞ through the vectorial Eikonal equation
[Contre-exemples dans le calcul des variations dans L^∞ par l'équation iconale vectorielle]

Présenté par : Haim Brezis

Katzourakis, Nikos ¹ ; Shaw, Giles

¹ Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, Berkshire, England, UK

Comptes Rendus. Mathématique, Tome 356 (2018) no. 5, pp. 498-502.

Résumé
Abstract

Nous montrons que, pour tout domaine borné régulier $Ω \subseteq R^{n}$ , $n = 2, 3$ , il existe une infinité de difféomorphismes globaux qui sont solutions de l'équation iconale, égaux à l'identité sur ∂Ω. Nous donnons également des exemples explicites de telles cartes dans des domaines annulaires. Ceci implique que le système du type ∞-Laplacien apparaissant dans le calcul des variations vectoriel dans $L^{\infty}$ ne suffit pas à caractériser les limites pour $p \to \infty$ des cartes p-harmoniques, ni les minimiseurs absolus au sens d'Aronsson.

We show that, for any regular bounded domain $Ω \subseteq R^{n}$ , $n = 2, 3$ , there exist infinitely many global diffeomorphisms equal to the identity on ∂Ω that solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the ∞-Laplace system arising in vectorial calculus of variations in $L^{\infty}$ does not suffice to characterise either limits of p-Harmonic maps as $p \to \infty$ or absolute minimisers in the sense of Aronsson.

Reçu le : 2018-01-29
Accepté le : 2018-04-12
Publié le : 2018-04-17

DOI : 10.1016/j.crma.2018.04.010

Affiliations des auteurs :

Katzourakis, Nikos ¹ ; Shaw, Giles

¹ Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, Berkshire, England, UK

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Katzourakis, Nikos; Shaw, Giles. Counterexamples in calculus of variations in L^∞ through the vectorial Eikonal equation. Comptes Rendus. Mathématique, Tome 356 (2018) no. 5, pp. 498-502. doi : 10.1016/j.crma.2018.04.010. http://www.numdam.org/articles/10.1016/j.crma.2018.04.010/

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