Energy-driven pattern formation

International Congress of Mathematicians, ICM 2006

Volumn 1, Issue , 2006, Pages 359-383

  Kohn, Robert V ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

Action minimization; Aviles giga problem; Calculus of variations; Cross tie wall; Martensitic transformation; Micromagnetics; Microstructure

Indexed keywords

AVILES-GIGA PROBLEM; CALCULUS OF VARIATIONS; CROSS-TIE WALL; GINZBURG-LANDAU MODELS; MARTENSITIC PHASE TRANSFORMATIONS; MICROMAGNETICS; NON-CONVEX VARIATIONAL PROBLEMS; UPPER AND LOWER BOUNDS;

MARTENSITIC TRANSFORMATIONS;

MICROSTRUCTURE;

EID: 84878064219     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (65)

1. Alouges F., Rivière, T., Serfaty, S., Néel and cross-tie wall energies for planar magnetic configurations. ESAIM Control Optim. Calc. Var. 8 (2002), 31-68.
2. Ambrosio, L., De Lellis, C., Mantegazza, C., Line energies for gradient vector fields in the plane. Calc. Var. PDE, 9 (1999), 327-355.
3. Ambrosio, L., Lecumberry, M., Rivière, T., A viscosity property of minimizing micromagnetic configurations. Comm. Pure Appl. Math. 56 (2003), 681-688.
4. Aviles, P., Giga, Y., A mathematical problem related to the physical theory of liquid crystal configurations. In Miniconference on geometry and partial differential equations. 2 (J. Hutchinson and L. Simon, eds.),.
5. Aviles, P., Giga, Y., Proc. Centre Math. Anal. Austral. Nat. Univ. 12, Australian National University, Centre for Mathematical Analysis, Canberra 1987, 1-16.
6. Aviles, P., Giga, Y., The distance function and defect energy. Proc. Roy. Soc. Edinburgh 126A (1996), 923-938.
7. Aviles, P., Giga, Y., On lower semicontinuity of a defect obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields. Proc. Roy. Soc. Edinburgh 129A (1999) 1-17.
8. Ben Belgacem, H., Conti, S., DeSimone, A., Müller, S., Energy scaling of compressed elastic films - three-dimensional elasticity and reduced theories. Arch. Rational Mech. Anal. 164 (2002), 1-37.
9. Bhattacharya, K., Microstructure of Martensite: Why it forms and how it gives rise to the shape-memory effect. Oxford Series on Materials Modelling, Oxford University Press, Oxford 2003.
10. Braides, A., I-Convergence for Beginners. Oxford University Press, Oxford 2002.
11. Casten, R. G., Holland, C. J., Instability results for reaction diffusion equations with Neumann boundary conditions. J. Differential Equations 27 (1978), 266-273.
12. Choksi, R., Scaling laws in microphase separation of diblock copolymers. J. Nonlinear Sci. 11 (2001), 223-236.
13. Choksi, R., Kohn, R. V., Bounds on the micromagnetic energy of a uniaxial ferromagnet. Comm. Pure Appl. Math. 51 (1998), 259-289.
14. Choksi, R., Kohn, R. V., Otto, F., Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Comm. Math. Phys. 201 (1999), 61-79.
15. Choksi, R., Kohn, R. V., Otto, F., Energy minimization and flux domain structure in the intermediate state of a type-I superconductor. J. Nonlinear Sci. 14 (2004), 119-171.
16. Conti, S., Branched microstructures: scaling and asymptotic self-similarity. Comm. Pure Appl. Math. 53 (2000), 1448-1474.
17. Conti, S., Niethammer, B., Otto, F., Coarsening rates in off-critical mixtures. SIAM J.Math. Anal. 37 (2006), 1732-1741.
18. Conti, S., Theil, F., Single-slip elastoplastic microstructures. Arch. Rational Mech. Anal. 178 (2005), 125-148.
19. DeLellis, C., Otto, F., Structure of entropy solutions to the eikonal equation. J. Eur. Math. Soc. 5 (2003), 107-145.
20. DeSimone, A., Kohn, R.V., Müller, S., Otto, F., A compactness result in the gradient theory of phase transitions. Proc. Roy. Soc. Edinburgh 131A (2001), 833-844.
21. DeSimone, A., Kohn, R. V., Müller, S., Otto, F., Repulsive interaction of Néel walls, and the internal length scale of the cross-tie wall. Multiscale Model. Simul. 1 (2003), 57-104.
22. DeSimone, A., Kohn, R. V., Müller, S., Otto, F., Recent analytical developments in micromagnetics. In The Science of Hysteresis II: Physical Modeling, Micromagnetics, and Magnetization Dynamics (G. Bertotti and I. Mayergoyz, eds.), Elsevier, 2006, 269-381.
23. E, W., Ren, W., Van Den Eijnden, E., String method for the study of rare events. Phys. Rev. B 66 (2002), 052301.
24. E, W., Ren, W., Van Den Eijnden, E., Energy landscape and thermally activated switching of submicron-size ferromagnetic elements. J. Appl. Phys. 93 (2003), 2275-2282.
25. E, W., Ren, W., Van Den Eijnden, E., Minimum action method for the study of rare events. Comm. Pure Appl. Math 57 (2004), 637-656.
26. E, W., Ren, W., Van Den Eijnden, E., Finite temperature string method for the study of rare events. J. Chem. Phys. 109B (2005), 6688-6693.
27. E, W., Ren, W., Van Den Eijnden, E., Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces, and transition tubes. Chem. Phys. Lett. 413 (2005), 242-247.
28. Ercolani, N., Indik, R., Newell, A. C., Passot, T., The geometry of the phase diffusion equation. J. Nonlinear Sci. 10 (2000), 223-274.
29. Faris, W. G., Jona-Lasinio, G., Large fluctuations for a nonlinear heat equation with noise. J. Phys. A: Math. Gen. 15 (1982), 3025-3055.
30. Fogedby, H. C., Hertz, J., Svane, A., Domainwall propagation and nucleation in a two-level system. Phys. Rev. E 70 (2004), 031105.
31. Freidlin, M. L., Wentzell, A. D., Random Perturbations of Dynamical Systems. Second edition, Grundlehren Math.Wiss. 260, Springer-Verlag, NewYork 1998.
32. Funaki, T., The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Related Fields 102 (1995), 221-288.
33. Gardiner, C.W., Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Third edition, Springer Ser. Synergetics 13, Springer-Verlag, Berlin 2004.
34. Hubert, A., Zur Theorie der zweiphasigen Domänenstrukturen in Supraleitern und Ferromagneten. Phys. Stat. Solidi 24 (1967), 669-682.
35. Hubert, A., R. Schäfer, R., Magnetic Domains: The Analysis of Magnetic Microstructures. Springer-Verlag, 1998.
36. Hutchinson, J., Tonegawa, Y., Convergence of phase interfaces in the van derWaals-Cahn- Hilliard theory. Calc. Var. PDE 10 (2000), 49-84.
37. Jabin, P. E., Perthame, B., Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001), 1096-1109.
38. Jabin, P. E., Perthame, B., Otto, F., Line-energy Ginzburg-Landau models: zero-energy states. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), 187-202.
39. Jin, W., Kohn, R. V., Singular perturbation and the energy of folds. J. Nonlinear Sci. 10 (2000), 355-390.
40. Kohn, R.V., Müller, S., Branching of twins near an austenite-twinned-martensite interface. Phil. Mag. A 66 (1992), 697-715.
41. Kohn, R. V., Müller, S., Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math. 47 (1994), 405-435.
42. Kohn, R. V., Otto, F., Upper bounds on coarsening rates. Comm. Math. Phys. 229 (2002), 375-395.
43. Kohn, R. V., Otto, F., Reznikoff, M. G., Van Den Eijnden, E., Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation Comm. Pure Appl. Math. 60 (2007), 393-438.
44. Kohn, R. V., Reznikoff, M.G., Tonegawa, Y., The sharp-interface limit of the action functional for Allen-Cahn in one space dimension. Calc. Var. PDE 25 (2006), 503-534.
45. Kohn, R. V., Yan, X., Coarsening rates for models of multicomponent phase separation. Interfaces Free Bound. 6 (2004) 135-149.
46. Lecumberry, M., Rivière, T., Regularity for micromagnetic configurations having zero jump energy. Calc. Var. PDE 15 (2002) 389-402.
47. Modica, L., Mortola, S., Un esempio di I-convergenza. Boll. Un. Mat. Ital. 14 (1977), 285-299.
48. Moser, R., A higher-order asymptotic problem related to phase transitions. SIAM J. Math. Anal. 37 (2005) 712-736.
49. Müller, S., Singular perturbation as a selection mechanism for periodic minimizing sequences. Calc. Var. PDE 1 (1993), 169-204.
50. Nagase, Y., Tonegawa, Y., A singular perturbation problem with integral curvature bound. Preprint.
51. Nakatani, Y., Uesaka, Y., Hayashi, N., Direct solution for the Landau-Lifshitz-Gilbert equation for micromagnetics. Japan J. Appl. Phys. 28 (1989), 2485-2507.
52. Otto, F., Cross-over in scaling laws: a simple example from micromagnetics. Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol. III, Higher Ed. Press, Beijing 2002, 829-838.
53. Privorotskii, I. Thermodynamics of Domain Structures. JohnWiley & Sons, 1976.
54. Ren, W., Higher order string method for finding minimum energy paths. Comm. Math. Sci. 1 (2003), 377-384.
55. Ren, X., Truskinovsky, L., Finite scale microstructures in nonlocal elasticity. J. Elasticity 59 (2000), 319-355.
56. Ren, X., Wei, J., On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5 (2003), 193-238.
57. Rivière, T., Serfaty, S., Limiting domain wall energy for a problem related to micromagnetics. Comm. Pure Appl. Math. 54 (2001), 294-338.
58. Rivière, T., Serfaty, S., Compactness, kinetic formulation, and entropies for a problem related to micromagnetics, Comm. Partial Differential Equations 28 (2003), 249-269.
59. Röger, M., Schätzle, R., On a modified conjecture of DeGiorgi. Preprint.
60. Sandier, E., Serfaty, S., Vortices in theMagnetic Ginzburg-Landau Model. Progr.Nonlinear Differential Equations Appl. 70, Birkhäuser, Boston 2007.
61. Serfaty, S., personal communication.
62. Tonegawa, Y., Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh 132A (2002), 993-1019.
63. Truskinovsky, L., Zanzotto, G., Ericksen's bar revisited: energy wiggles. J. Mech. Phys. Solids 44 (1996), 1371-1408.
64. van den Berg, H. A. M. Self-consistent domain theory in soft-ferromagnetic media. II. Basic domain structures in thin film objects. J. Appl. Phys. 60 (1986), 1104-1113.
65. Yip, N. K., Structure of stable solutions of a one-dimensional variational problem. ESAIM Control Optim. Calc. Var. 12 (2006), 721-751.