A diffused interface whose chemical potential lies in a Sobolev space
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 4 (2005) no. 3, p. 487-510
We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms W 1,p of the associated chemical potential fields are bounded uniformly, where p>n 2 and n is the dimension of the domain. We show that the limit interface as ε tends to zero is an integral varifold with a sharp integrability condition on the mean curvature.
Classification:  35J60,  35B25,  35J20,  80A22
@article{ASNSP_2005_5_4_3_487_0,
     author = {Tonegawa, Yoshihiro},
     title = {A diffused interface whose chemical potential lies in a Sobolev space},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {3},
     year = {2005},
     pages = {487-510},
     zbl = {1170.35416},
     mrnumber = {2185866},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2005_5_4_3_487_0}
}
Tonegawa, Yoshihiro. A diffused interface whose chemical potential lies in a Sobolev space. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 4 (2005) no. 3, pp. 487-510. http://www.numdam.org/item/ASNSP_2005_5_4_3_487_0/

[1] W. Allard, On the first variation of a varifold, Ann. of Math. 95 (1972), 417-491. | MR 307015 | Zbl 0252.49028

[2] G. Bellettini and L. Mugnai, On the approximation of the elastica functional in radial symmetry, Calc. Var. 24 (2005), 1-20. | MR 2157848 | Zbl 1086.49009

[3] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258-267.

[4] X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Differential Geom. 44 (1996), 262-141. | MR 1425577 | Zbl 0874.35045

[5] Q. Du, C. Liu, R. Ryham and X. Wang, The phase field formulation of the Willmore problem, Nonlinearity 18 (2005), 1249-1267. | MR 2134893 | Zbl 1125.35366

[6] L. C. Evans and R. F. Gariepy, “Measure theory and fine properties of functions”, Studies in Advanced Math., CRC Press, 1992. | MR 1158660 | Zbl 0804.28001

[7] D. Gilbarg and N. S. Trudinger, “Elliptic partial differential equations of second order”, 2nd Edition, Springer-Verlag, 1983. | MR 737190 | Zbl 0562.35001

[8] M. E. Gurtin, Some results and conjectures in the gradient theory of phase transitions, In: “Metastability and incompletely posed problems”, S. Antman et al. (eds.), pp. 301-317, Springer, 1987. | MR 870014 | Zbl 0634.49019

[9] J. E. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der Waals - Cahn - Hilliard theory, Calc. Var. 10 (2000), 49-84. | MR 1803974 | Zbl 1070.49026

[10] R. Kohn, M. Reznikoff and Y. Tonegawa, The sharp-interface limit of the action functional for Allen Cahn in one space dimension, to appear in Cal. Var. | Zbl 1094.60044

[11] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal. 98 (1987), 123-142. | MR 866718 | Zbl 0616.76004

[12] R. Moser, A higher order asymptotic problem related to phase transitions, preprint. | MR 2191773

[13] Y. Nagase and Y. Tonegawa, A singular perturbation problem with a Willmore-type energy bound, in preparation.

[14] P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Comm. Pure Appl. Math. 51 (1998), 551-579. | MR 1611144 | Zbl 0955.58011

[15] M. Röger and R. Schätzle, On a modified conjecture of De Giorgi, in preparation.

[16] R. Schätzle, Hypersurfaces with mean curvature given by an ambient Sobolev function, J. Diffefential Geom. 58 (2001), 371-420. | MR 1906780 | Zbl 1055.49032

[17] L. Simon, “Lectures on geometric measure theory”, Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 3, 1983. | MR 756417 | Zbl 0546.49019

[18] P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Ration. Mech. Anal. 101 (1988), no. 3, 209-260. | MR 930124 | Zbl 0647.49021

[19] Y. Tonegawa, Phase field model with a variable chemical potential, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), 993-1019. | MR 1926927 | Zbl 1013.35070