Curvature theory of boundary phases: The two-dimensional case

Interfaces and Free Boundaries

Volumn 4, Issue 4, 2002, Pages 345-370

  Braides, Andrea ^a   Malchiodi, Andrea ^b  

^a UNIVERSITY OF ROME TOR VERGATA   (Italy)

^b INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

Curvature functionals; G convergence; Non convex problems; Phase transitions; Surface energies

Indexed keywords

EID: 30544433554     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/65     Document Type: Article

References (30)

1. ALBERTI, G. & BELLETTINI, G. A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies. European J. Appl. Math. 9, (1998) 261-284.
2. ALBERTI, G. & BELLETTINI, G. A nonlocal anisotropic model for phase transitions. I. The optimal profile problem. Math. Ann. 310, (1998) 527-560.
3. ALBERTI, G., BELLETTINI, G., CASSANDRO, M., & PRESUTTI, E. Surface tension in Ising systems with Kac potentials. J. Statist. Phys. 82, (1996) 743-796.
4. AMBROSIO, L. & BRAIDES, A. Functionals defined on partitions in sets of finite perimeter. II. Semicontinuity, relaxation and homogenization. J. Math. Pures Appl. 69, (1990) 307-333.
5. AMBROSIO, L., FUSCO, N., & PALLARA, D. Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000).
6. BELLETTINI, G., DAL MASO, G., & PAOLINI, M. Semicontinuity and relaxation properties of a curvature depending functional in 2D. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20, (1993) 247-297.
7. BELLETTINI, G., GOGLIONE, R., & NOVAGA, M. Approximation to driven motion by crystalline curvature in two dimensions. Adv. Math. Sci. Appl. 10, (2000) 467-493.
8. BODINEAU, T. The Wulff construction in three and more dimensions. Comm. Math. Phys. 207, (1999) 197-229.
9. BODINEAU, T., IOFFE, D., & VELENIK, Y. Rigorous probabilistic analysis of equilibrium crystal shapes. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41, (2000) 1033-1098.
10. BRAIDES, A. Approximation of Free-Discontinuity Problems. Springer, Berlin (1998).
11. BRAIDES, A. G-convergence for Beginners. Oxford University Press, Oxford (2002).
12. COSCIA, A. On curvature sensitive image segmentation. Nonlinear Anal. 39, (2000) 711-730.
13. DAL MASO, G. An Introduction to G-convergence. Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser, Boston (1993).
14. DAL MASO, G. & ROMANELLI, M. Paper in preparation.
15. FONSECA, I. Variational methods for elastic crystals. Arch. Rational Mech. Anal. 97, (1987) 189-220.
16. FONSECA, I. The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67, (1988) 175-195.
17. GIGA, Y., GURTIN, M. E., & MATIAS, J. On the dynamics of crystalline motions. Japan J. Indust. Appl. Math. 15, (1998) 7-50.
18. GIUSTI, E. Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics 80. Birkhäuser, Boston, MA (1984).
19. GOGLIONE, R. & PAOLINI, M. Numerical simulations of crystalline motion by mean curvature with Allen-Cahn relaxation. Free Boundary Problems, Theory and Applications (Zakopane, 1995), Pitman Res. Notes Math. Ser. vol. 363. Longman, Harlow (1996) pp. 203-216.
20. GURTIN, M. E. Some results and conjectures in the gradient theory of phase transitions, in "Metastability and incompletely posed problems, Proc. Workshop, Minneapolis/Minn. 1984/85". IMA Vol. Math. Appl. 3, (1987) 135-146.
21. GURTIN, M. E., SONER, H. M., & SOUGANIDIS, P. E. Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles. J. Differential Equations 119, (1995) 54-108.
22. LUCKHAUS, S. & MODICA, L. The Gibbs-Thompson relation within the gradient theory of phase transitions. Arch. Rational Mech. Anal. 107, (1989) 71-83.
23. MANTEGAZZA, C. Curvature varifolds with boundary. J. Differential Geom. 43, (1996) 807-843.
24. MODICA, L. The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98, (1987) 123-142.
25. MODICA, L. & MORTOLA, S. Un esempio di G -convergenza. Boll. Un. Mat. Ital. B 14, (1977) 285-299.
26. MUMFORD, D. Elastica and computer vision. Algebraic Geometry and its Applications (West Lafayette, IN, 1990). Springer, New York (1994) pp. 491-506.
27. NITZBERG, M., MUMFORD, D., & SHIOTA, T. Filtering, Segmentation and Depth. Lecture Notes in Computer Science, 662. Springer, Berlin (1993).
28. TAYLOR, J. E. On the global structure of crystalline surfaces. Discrete Comput. Geom. 6, (1991) 225-262.
29. TAYLOR, J. E. Constructions and conjectures in crystalline nondifferential geometry. Differential Geometry, Pitman Monographs Surveys Pure Appl. Math. vol. 52. Longman, Harlow (1991) pp. 321-336.
30. TAYLOR, J. E. & CAHN, J. W. Diffuse interfaces with sharp corners and facets: phase field models with strongly anisotropic surfaces. With an appendix by Jason Yunger. Phys. D 112, (1998) 381-411.