A higher order asymptotic problem related to phase transitions

SIAM Journal on Mathematical Analysis

Volumn 37, Issue 3, 2006, Pages 712-736

  Moser, Roger ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

Generalized surfaces; Phase transitions; Willmore functional

Indexed keywords

GENERALIZED SURFACES; MONOTONICITY; WAALS-CAHNHILLIARD THEORY; WILLMORE FUNCTIONAL;

ASYMPTOTIC STABILITY; FUNCTIONS; NUMERICAL METHODS; PHASE TRANSITIONS;

PROBLEM SOLVING;

EID: 33644553274     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/040616760     Document Type: Article

References (24)

1. W. K. ALLARD. On the first variation of a varifold, Ann. of Math. (2), 95 (1972), pp. 417-491.
2. 3 and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), pp. 725-739.
3. L. AMBROSIO, N. FUSCO, AND D. PALLARA, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York. 2000.
4. L. AMBROSIO AND H. M. SONER, A measure-theoretic approach to higher codimension mean curvature flows, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), pp. 27-49.
5. G. BELLETTINI AND L. MUGNAI, On the approximation of the elastica functional in radial symmetry, Calc. Var. Partial Differential Equations, 24 (2005), pp. 1-20.
6. E. DE GIORGI, Some remarks on Γ-convergence and least squares method, in Composite Media and Homogenization Theory, Trieste, 1990, Progr. Nonlinear Differential Equations Appl. 5, Birkhäuser Boston, Boston, 1991, pp. 135-142.
7. H. FEDERER. Geometric Measure Theory, Springer-Verlag, New York, 1969.
8. K. GIUSTI, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser-Verlag, Basel, 1984.
9. J. E. HUTCHINSON AND Y. TONEGAWA, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory, Calc. Var. Partial Differential Equations, 10 (2000), pp. 49-84.
10. F.-H. LIN, Varifold type theory for Sobolev mappings, in First International Congress of Chinese Mathematicians (Beijing, 1998), AMS, Providence, RI, 2001, pp. 423-430.
11. L. MODICA, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), pp. 679-684.
12. L. MODICA, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), pp. 123-142.
13. L. MODICA, Gradient theory of phase transitions with boundary contact energy, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), pp. 487-512.
14. L. MODICA AND S. MORTOLA, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici, Boll. Un. Mat. Ital. A (5), 14 (1977), pp. 526-529.
15. L. MODICA AND S. MORTOLA, Un esempio di Γ-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), pp. 285-299.
16. R. MOSEH, Stationary measures and rectifiability, Calc. Var. Partial Differential Equations, 17 (2003), pp. 357-368.
17. R. MOSER, Energy concentration for almost harmonic maps and the Willmore functional, Math. Z., 251 (2005), pp. 293-311.
18. R. MOSER, On the structure of energy concentration measures, Calc. Var. Partial Differential Equations, to appear.
19. P. PRICE, A monotonicity formula for Yang-Mills fields, Manuscripta Math., 43 (1983), pp. 131-166.
20. L. SIMON, Lectures on Geometric Measure Theory, Australian National University Centre for Mathematical Analysis, Canberra, 1983.
21. L. SIMON, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), pp. 281-326.
22. P. STERNBERG, The effect of a singular perturbation on nonconvex variational problems, Arch. Ration. Mech. Anal., 101 (1988), pp. 209-260.
23. Y. TONEGAWA, Phase field model with a variable chemical potential, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), pp. 993-1019.
24. T. J. WILLMORE, Riemannian Geometry, Oxford Sci. Pub., The Clarendon Press, Oxford University Press, New York, 1993.