Partial differential equations/Calculus of variations
Counterexamples in calculus of variations in L through the vectorial Eikonal equation
Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 498-502.

We show that, for any regular bounded domain ΩRn, 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 does not suffice to characterise either limits of p-Harmonic maps as p or absolute minimisers in the sense of Aronsson.

Nous montrons que, pour tout domaine borné régulier ΩRn, 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 ne suffit pas à caractériser les limites pour p des cartes p-harmoniques, ni les minimiseurs absolus au sens d'Aronsson.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.010
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, Volume 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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