Partial differential equations
Explicit 2D ∞-harmonic maps whose interfaces have junctions and corners
Comptes Rendus. Mathématique, Volume 351 (2013) no. 17-18, pp. 677-680.

Given a map u:ΩRnRN, the ∞-Laplacian is the system:

Δu:=(DuDu+|Du|2[Du]I):D2u=0(1)
and arises as the “Euler–Lagrange PDE” of the supremal functional E(u,Ω)=DuL(Ω). (1) is the model PDE of the vector-valued Calculus of Variations in L 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.

On se donne une carte u:ΩRnRN, le laplacien-∞ est le système :

Δu:=(DuDu+|Du|2[Du]I):D2u=0,(1)
qui se présente comme une EDP dʼEuler–Lagrange de la fonctionnelle E(u,Ω)=DuL(Ω) ; (1) est lʼEDP modèle du calcul des variations à valeurs vectorielles dans L, 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.07.028
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
@article{CRMATH_2013__351_17-18_677_0,
     author = {Katzourakis, Nicholas},
     title = {Explicit {2\protect\emph{D}} \ensuremath{\infty}-harmonic maps whose interfaces have junctions and corners},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {677--680},
     publisher = {Elsevier},
     volume = {351},
     number = {17-18},
     year = {2013},
     doi = {10.1016/j.crma.2013.07.028},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.07.028/}
}
TY  - JOUR
AU  - Katzourakis, Nicholas
TI  - Explicit 2D ∞-harmonic maps whose interfaces have junctions and corners
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 677
EP  - 680
VL  - 351
IS  - 17-18
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.07.028/
DO  - 10.1016/j.crma.2013.07.028
LA  - en
ID  - CRMATH_2013__351_17-18_677_0
ER  - 
%0 Journal Article
%A Katzourakis, Nicholas
%T Explicit 2D ∞-harmonic maps whose interfaces have junctions and corners
%J Comptes Rendus. Mathématique
%D 2013
%P 677-680
%V 351
%N 17-18
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.07.028/
%R 10.1016/j.crma.2013.07.028
%G en
%F CRMATH_2013__351_17-18_677_0
Katzourakis, Nicholas. Explicit 2D ∞-harmonic maps whose interfaces have junctions and corners. Comptes Rendus. Mathématique, Volume 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/

[1] Aronsson, G. Extension of functions satisfying Lipschitz conditions, Ark. Mat., Volume 6 (1967), pp. 551-561

[2] Aronsson, G. On the partial differential equation ux2uxx+2uxuyuxy+uy2uyy=0, Ark. Mat., Volume 7 (1968), pp. 395-425

[3] Aronsson, G. On certain singular solutions of the partial differential equation ux2uxx+2uxuyuxy+uy2uyy=0, Manuscr. Math., Volume 47 (1984) no. 1–3, pp. 133-151

[4] Aronsson, G. Construction of singular solutions to the p-harmonic equation and its limit equation for p=, Manuscr. Math., Volume 56 (1986), pp. 135-158

[5] Barron, N. Viscosity solutions and analysis in L, Montreal, QC, 1998, Kluwer Acad. Publ., Dordrecht (1999), pp. 1-60

[6] Capogna, L.; Raich, A. An Aronsson-type approach to extremal quasiconformal mappings, J. Differ. Equ., Volume 253 (2012) no. 3, pp. 851-877

[7] Crandall, M.G. A visit with the ∞-Laplacian, Calculus of Variations and Non-Linear PDE, Springer Lecture Notes in Mathematics, vol. 1927, CIME, Cetraro, Italy, 2005

[8] Crandall, M.G.; Ishii, H.; Lions, P.-L. Userʼs guide to viscosity solutions of 2nd-order partial differential equations, Bull. Am. Math. Soc., Volume 27 (1992) no. 1, pp. 1-67

[9] Katzourakis, N. Explicit singular viscosity solutions of the Aronsson equation, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 21–22, pp. 1173-1176

[10] Katzourakis, N. L variational problems for maps and the Aronsson PDE system, J. Differ. Equ., Volume 253 (2012) no. 7, pp. 2123-2139

[11] N. Katzourakis, On the structure of ∞-harmonic maps, preprint, 2012.

[12] N. Katzourakis, Optimal ∞-quasiconformal immersions, preprint, 2012.

[13] Katzourakis, N. The subelliptic ∞-Laplace system on Carnot–Carathéodory spaces, Adv. Nonlinear Anal., Volume 2 (2013) no. 2, pp. 213-233

[14] Katzourakis, N. ∞-Minimal submanifolds, Proceedings of the AMS, 2013 (in press)

[15] Sheffield, S.; Smart, C.K. Vector valued optimal Lipschitz extensions, Commun. Pure Appl. Math., Volume 65 (2012) no. 1, pp. 128-154

Cited by Sources: