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

Nikos Katzourakis; Giles Shaw

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

Presented by: Haim Brezis

Nikos Katzourakis ¹; Giles Shaw

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

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

Abstract
Résumé

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.

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.

Received: 2018-01-29
Accepted: 2018-04-12
Published online: 2018-04-17

DOI: 10.1016/j.crma.2018.04.010

Author's affiliations:

Nikos Katzourakis ¹; Giles Shaw

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

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     author = {Nikos Katzourakis and Giles Shaw},
     title = {Counterexamples in calculus of variations in {\protect\emph{L}\protect\textsuperscript{\ensuremath{\infty}}} through the vectorial {Eikonal} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {498--502},
     publisher = {Elsevier},
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     number = {5},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.010},
     language = {en},
}

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Nikos Katzourakis; Giles Shaw. Counterexamples in calculus of variations in L^∞ through the vectorial Eikonal equation. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 498-502. doi : 10.1016/j.crma.2018.04.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.010/

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