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 & 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, ().   Google Scholar

[2]

E. Bodenschatz, W. Pesch and G. Ahlers, Recent developments in Rayleigh-Bénard convection,, in, 32 (2000), 709.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[5]

N. Ercolani, R. Indik, A. C. Newell and T. Passot, Global description of patterns far from onset: A case study,, in, 1 (2003), 411.   Google Scholar

[6]

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

[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.   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

show all references

References:
[1]

J. M. Ball and A. Zarnescu, Orientability and energy minimization in liquid crystal models,, preprint, ().   Google Scholar

[2]

E. Bodenschatz, W. Pesch and G. Ahlers, Recent developments in Rayleigh-Bénard convection,, in, 32 (2000), 709.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[5]

N. Ercolani, R. Indik, A. C. Newell and T. Passot, Global description of patterns far from onset: A case study,, in, 1 (2003), 411.   Google Scholar

[6]

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

[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.   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[1]

Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170

[2]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260

[3]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145

[4]

Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162

[5]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388

[6]

Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021001

[7]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176

[8]

Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic & Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048

[9]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045

[10]

Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390

[11]

Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020366

[12]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[13]

Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1

[14]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[15]

Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097

[16]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[17]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021021

[18]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[19]

Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367

[20]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]