Explicit 2<i>D</i> ∞-harmonic maps whose interfaces have junctions and corners

Katzourakis, Nicholas

doi:10.1016/j.crma.2013.07.028

Partial differential equations

Explicit 2D ∞-harmonic maps whose interfaces have junctions and corners
[Applications harmoniques-∞ explicites bidimensionnelles présentant des jonctions et des coins]

Présenté par : Haim Brezis

Katzourakis, Nicholas ^1,²

¹ BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E-48009, Bilbao, Spain
² Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, Berkshire, UK

Comptes Rendus. Mathématique, Tome 351 (2013) no. 17-18, pp. 677-680.

Résumé
Abstract

On se donne une carte $u : Ω \subseteq R^{n} \to R^{N}$ , le laplacien-∞ est le système :

Δ_{\infty} u : = (D u \otimes D u + {| D u |}^{2} {[D u]}^{⊥} \otimes I) : D^{2} u = 0,

(1)

qui se présente comme une EDP dʼEuler–Lagrange de la fonctionnelle

E_{\infty} (u, Ω) = {‖ D u ‖}_{L^{\infty} (Ω)}

; (1) est lʼEDP modèle du calcul des variations à valeurs vectorielles dans

L^{\infty}

, introduite pour la première fois dans les travaux de lʼauteur [10–14]. Les solutions de (1) mettent en évidence une séparation naturelle, avec des comportements qualitativement différents pour chaque phase. De plus, sur les interfaces, les coefficients de (1) sont discontinus. On construit ici des solutions régulières explicites dans le cas

n = N = 2

, solutions pour lesquelles des jonctions ont des points triples et des coins non réguliers. Lʼextrême complexité de ces solutions permet de mieux comprendre lʼEDP (1) et ses limites, qui pourraient être vraies pour dʼautres cas envisageables de régularité des interfaces.

Given a map $u : Ω \subseteq R^{n} \to R^{N}$ , the ∞-Laplacian is the system:

Δ_{\infty} u : = (D u \otimes D u + {| D u |}^{2} {[D u]}^{⊥} \otimes I) : D^{2} u = 0

(1)

and arises as the “Euler–Lagrange PDE” of the supremal functional

E_{\infty} (u, Ω) = {‖ D u ‖}_{L^{\infty} (Ω)}

. (1) is the model PDE of the vector-valued Calculus of Variations in

L^{\infty}

and first appeared in the authorʼs recent work [10–14]. Solutions to (1) present a natural phase separation with qualitatively different behaviour on each phase. Moreover, on the interfaces the coefficients of (1) are discontinuous. Herein we construct new explicit smooth solutions for

n = N = 2

, for which the interfaces have triple junctions and non-smooth corners. The high complexity of these solutions provides further understanding of the PDE (1) and limits what might be true in future regularity considerations of the interfaces.

Reçu le : 2013-03-07
Accepté le : 2013-07-31
Publié le : 2013-08-28

DOI : 10.1016/j.crma.2013.07.028

Affiliations des auteurs :

Katzourakis, Nicholas ^{1, 2}

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     title = {Explicit {2\protect\emph{D}} \ensuremath{\infty}-harmonic maps whose interfaces have junctions and corners},
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Katzourakis, Nicholas. Explicit 2D ∞-harmonic maps whose interfaces have junctions and corners. Comptes Rendus. Mathématique, Tome 351 (2013) no. 17-18, pp. 677-680. doi : 10.1016/j.crma.2013.07.028. http://www.numdam.org/articles/10.1016/j.crma.2013.07.028/

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