February  2012, 5(1): 183-189. doi: 10.3934/dcdss.2012.5.183

Stripe patterns and the Eikonal equation

1. 

Dept. of Mathematics and Computer Science and Institute for Complex Molecular Systems, Technische Universiteit Eindhoven, PO Box 513, 5600 MB Eindhoven, Netherlands

2. 

Technische Universität Dortmund, Fakultät für Mathematik, Lehrstuhl I, Vogelpothsweg 87, 44227 Dortmund, Germany

Received  April 2009 Revised  December 2009 Published  February 2011

We study a new formulation for the Eikonal equation $|\nabla u| =1$ on a bounded subset of $\R^2$. Considering a field $P$ of orthogonal projections onto $1$-dimensional subspaces, with div$ P \in L^2$, we prove existence and uniqueness for solutions of the equation $P$ div $P$=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve.
   This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals.
Citation: Mark A. Peletier, Marco Veneroni. Stripe patterns and the Eikonal equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 183-189. doi: 10.3934/dcdss.2012.5.183
References:
[1]

J. M. Ball and A. Zarnescu, Orientability and energy minimization in liquid crystal models, preprint, arXiv:1009.2688.

[2]

E. Bodenschatz, W. Pesch and G. Ahlers, Recent developments in Rayleigh-Bénard convection, in "Annual Review of Fluid Mechanics," volume 32 of Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, (2000), 709-778.

[3]

J. A. Boon, C. J. Budd and G. W. Hunt, Level set methods for the displacement of layered materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1447-1466.

[4]

M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.

[5]

N. Ercolani, R. Indik, A. C. Newell and T. Passot, Global description of patterns far from onset: A case study, in "Nonlinear Dynamics (Canberra, 2002),'' volume 1 of World Sci. Lect. Notes Complex Syst., World Sci. Publ., River Edge, NJ, (2003), 411-435.

[6]

J. E. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J., 35 (1986), 45-71. doi: 10.1512/iumj.1986.35.35003.

[7]

P.-E. Jabin, F. Otto and B. Perthame, Line-energy Ginzburg-Landau models: Zero-energy states, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 1 (2002), 187-202.

[8]

M. Peletier and M. Röger, Partial localization, lipid bilayers, and the elastica functional, Arch. Ration. Mech. Anal., 193 (2009), 475-537. doi: 10.1007/s00205-008-0150-4.

[9]

M. A. Peletier and M. Veneroni, Non-oriented solutions of the eikonal equation, C. R. Math. Acad. Sci. Paris., 348 (2010), 1099-1101. doi: 10.1016/j.crma.2010.09.011.

[10]

M. A. Peletier and M. Veneroni, Stripe patterns in a model for block copolymers, Math. Models Methods Appl. Sci., 20 (2010), 843-907. doi: 10.1142/S0218202510004465.

[11]

A. Ruzette and L. Leibler, Block copolymers in tomorrow's plastics, Nature Materials, 4 (2005), 19-31. doi: 10.1038/nmat1295.

show all references

References:
[1]

J. M. Ball and A. Zarnescu, Orientability and energy minimization in liquid crystal models, preprint, arXiv:1009.2688.

[2]

E. Bodenschatz, W. Pesch and G. Ahlers, Recent developments in Rayleigh-Bénard convection, in "Annual Review of Fluid Mechanics," volume 32 of Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, (2000), 709-778.

[3]

J. A. Boon, C. J. Budd and G. W. Hunt, Level set methods for the displacement of layered materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1447-1466.

[4]

M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.

[5]

N. Ercolani, R. Indik, A. C. Newell and T. Passot, Global description of patterns far from onset: A case study, in "Nonlinear Dynamics (Canberra, 2002),'' volume 1 of World Sci. Lect. Notes Complex Syst., World Sci. Publ., River Edge, NJ, (2003), 411-435.

[6]

J. E. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J., 35 (1986), 45-71. doi: 10.1512/iumj.1986.35.35003.

[7]

P.-E. Jabin, F. Otto and B. Perthame, Line-energy Ginzburg-Landau models: Zero-energy states, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 1 (2002), 187-202.

[8]

M. Peletier and M. Röger, Partial localization, lipid bilayers, and the elastica functional, Arch. Ration. Mech. Anal., 193 (2009), 475-537. doi: 10.1007/s00205-008-0150-4.

[9]

M. A. Peletier and M. Veneroni, Non-oriented solutions of the eikonal equation, C. R. Math. Acad. Sci. Paris., 348 (2010), 1099-1101. doi: 10.1016/j.crma.2010.09.011.

[10]

M. A. Peletier and M. Veneroni, Stripe patterns in a model for block copolymers, Math. Models Methods Appl. Sci., 20 (2010), 843-907. doi: 10.1142/S0218202510004465.

[11]

A. Ruzette and L. Leibler, Block copolymers in tomorrow's plastics, Nature Materials, 4 (2005), 19-31. doi: 10.1038/nmat1295.

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