Nous étudions une nouvelle formulation de l'équation eikonale sur un sous-ensemble borné de . Au lieu d'un champ de vecteurs ∇u, nous considérons un champ P de projections orthogonales sur les sous-espaces de dimension 1, avec . Nous montrons que les solutions de cette équation propagent la direction comme dans l'équation eikonale classique. Nous montrons aussi que les solutions existent si et seulement si le domaine est un voisinage tubulaire d'une courbe régulière fermée.
We study a new formulation for the eikonal equation on a bounded subset of . Instead of a vector field ∇u, we consider a field P of orthogonal projections on one-dimensional subspaces, with . We prove that solutions of this equation propagate direction as in the classical eikonal equation. We also show that solutions exist if and only if the domain is a tubular neighborhood of a regular closed curve.
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@article{CRMATH_2010__348_19-20_1099_0, author = {Peletier, Mark A. and Veneroni, Marco}, title = {Non-oriented solutions of the eikonal equation}, journal = {Comptes Rendus. Math\'ematique}, pages = {1099--1101}, publisher = {Elsevier}, volume = {348}, number = {19-20}, year = {2010}, doi = {10.1016/j.crma.2010.09.011}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2010.09.011/} }
TY - JOUR AU - Peletier, Mark A. AU - Veneroni, Marco TI - Non-oriented solutions of the eikonal equation JO - Comptes Rendus. Mathématique PY - 2010 SP - 1099 EP - 1101 VL - 348 IS - 19-20 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2010.09.011/ DO - 10.1016/j.crma.2010.09.011 LA - en ID - CRMATH_2010__348_19-20_1099_0 ER -
%0 Journal Article %A Peletier, Mark A. %A Veneroni, Marco %T Non-oriented solutions of the eikonal equation %J Comptes Rendus. Mathématique %D 2010 %P 1099-1101 %V 348 %N 19-20 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2010.09.011/ %R 10.1016/j.crma.2010.09.011 %G en %F CRMATH_2010__348_19-20_1099_0
Peletier, Mark A.; Veneroni, Marco. Non-oriented solutions of the eikonal equation. Comptes Rendus. Mathématique, Tome 348 (2010) no. 19-20, pp. 1099-1101. doi : 10.1016/j.crma.2010.09.011. http://archive.numdam.org/articles/10.1016/j.crma.2010.09.011/
[1] Orientable and non-orientable director fields for liquid crystals, PAMM, Volume 7 (2007), pp. 1050701-1050704
[2] Global description of patterns far from onset: a case study, Canberra, 2002 (World Sci. Lect. Notes Complex Syst.), Volume vol. 1, World Sci. Publ., River Edge, NJ (2003), pp. 411-435
[3] Line-energy Ginzburg–Landau models: zero-energy states, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 1 (2002) no. 1, pp. 187-202
[4] Stripe patterns in a model for block copolymers, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 6, pp. 843-907
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