Abstract
We study the Γ-convergence of functionals arising in the Van der Waals–Cahn–Hilliard theory of phase transitions. Their limit is given as the sum of the area and the Willmore functional. The problem under investigation was proposed as modification of a conjecture of De Giorgi and partial results were obtained by several authors. We prove here the modified conjecture in space dimensions n = 2,3.
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References
Allard W.K. (1972) On the first variation of a varifold. Ann. Math. 95, 417–491
Bellettini G., Mugnai L. (2005) On the approximation of the elastica functional in radial symmetry. Cal. Var. Partial Differential Equations 24(1): 1–20
Bellettini G., Paolini, M. Approssimazione variazionale di funzionali con curvatura. Seminario di Analisi Matematica, Dipartimento di Matematica dell’Università di Bologna. Tecnoprint Bologna, 87–97 (1993)
Chen X. (1996) Global asymptotic limit of solutions of the Cahn–Hilliard equation. J Differential Geom. 44(2): 262–311
de Giorgi, E. Some remarks on Γ-convergence and least squares methods. In: Dal Maso, G., Dell’Antonio, G.F. (eds), Composite Media Homogenization Theory Progress in Nonlinear Differential Equations and their Applications, vol. 5, Birkhäuser, 135–142 (1991)
Gilbarg D., Trudinger N.S. (1998) Elliptic Partial Differential Equations of Second Order. Springer, Berlin Heidelberg New York
Hutchinson J., Tonegawa Y. (2000) Convergence of phase interfaces in the van der Waals–Cahn–Hilliard Theory. Cal. Var. Partial Differential Equations 10(1): 49–84
Loreti P., March R. (2000) Propagation of fronts in a nonlinear fourth order equation. Eur. J App. Math. 11, 203–213
Modica L., Mortola S. (1977) Un esempio di Γ-convergenza. Bollettino della Unione Matematica Italiana 14B, 285–299
Modica L. (1987) The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98(2): 123–142
Moser R. (2004) A higher order asymptotic problem related to phase transitions. SIAM J. Math. Anal. 37, 712–736
Padilla P., Tonegawa Y. (1998) On the convergence of stable phase transitions. Commun. Pure Appl. Math. 51(6): 551–579
Schätzle, R. Lower semicontinuity of the Willmore functional for currents (2004) (submitted)
Simon, L. Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis Australian National University, vol 3 (1983)
Tonegawa Y. (2002) Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh, Sect. A, 132(4): 993–1019
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This work was supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00274, FRONTS-SINGULARITIES.
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Röger, M., Schätzle, R. On a Modified Conjecture of De Giorgi. Math. Z. 254, 675–714 (2006). https://doi.org/10.1007/s00209-006-0002-6
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DOI: https://doi.org/10.1007/s00209-006-0002-6