We consider a mesoscopic model for phase transitions in a periodic medium and we construct multibump solutions. The rational perturbative case is dealt with by explicit asymptotics.
Mots clés : oscillatory solutions of PDEs, phase transitions, asymptotic expansions
@article{COCV_2009__15_4_914_0,
author = {Novaga, Matteo and Valdinoci, Enrico},
title = {Multibump solutions and asymptotic expansions for mesoscopic {Allen-Cahn} type equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {914--933},
publisher = {EDP-Sciences},
volume = {15},
number = {4},
year = {2009},
doi = {10.1051/cocv:2008058},
mrnumber = {2567252},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008058/}
}
TY - JOUR AU - Novaga, Matteo AU - Valdinoci, Enrico TI - Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 914 EP - 933 VL - 15 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008058/ DO - 10.1051/cocv:2008058 LA - en ID - COCV_2009__15_4_914_0 ER -
%0 Journal Article %A Novaga, Matteo %A Valdinoci, Enrico %T Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 914-933 %V 15 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008058/ %R 10.1051/cocv:2008058 %G en %F COCV_2009__15_4_914_0
Novaga, Matteo; Valdinoci, Enrico. Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 914-933. doi : 10.1051/cocv:2008058. http://www.numdam.org/articles/10.1051/cocv:2008058/
[1] , and , Existence of infinitely many stationary layered solutions in for a class of periodic Allen-Cahn equations. Comm. Partial Diff. Eq. 27 (2002) 1537-1574. | MR | Zbl
[2] and , A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1084-1095.
[3] and , Homoclinics: Poincaré-Melnikov type results via a variational approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 233-252. | Numdam | MR | Zbl
[4] , On the spectrum of the Schrödinger operator perturbed by a rapidly oscillating potential. J. Math. Sci. (N. Y.) 139 (2006) 6243-6322. | MR | Zbl
[5] , Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983). | MR | Zbl
[6] , Unicité et minimalité des solutions d'une équation de Ginzburg-Landau. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 305-318. | Numdam | MR | Zbl
[7] and , Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media. Adv. Math. 215 (2007) 379-426. | MR | Zbl
[8] and , Sharp-interface limit of a Ginzburg-Landau functional with a random external field. Preprint, http://www.mat.uniroma3.it/users/orlandi/pubb.html (2007). | MR
[9] and , Pinning and de-pinning phenomena in front propagation in heterogeneous media. Interfaces Free Bound. 8 (2006) 79-109. | MR | Zbl
[10] , and , -convergence of the Allen-Cahn energy with an oscillating forcing term. Interfaces Free Bound. 8 (2006) 47-78. | MR | Zbl
[11] , Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI (1998). | MR | Zbl
[12] and , Geometry of quasiminimal phase transitions. Calc. Var. Partial Differential Equations 33 (2008) 1-35. | MR | Zbl
[13] , The elements of mechanics, Texts and Monographs in Physics. Springer-Verlag, New York (1983). Translated from the Italian. | MR | Zbl
[14] and , Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften 224. Springer-Verlag, Berlin, second edition (1983). | MR | Zbl
[15] and , On the theory of superfluidity. Soviet Physics. JETP 34 (1958) 858-861 (Ž. Eksper. Teoret. Fiz. 1240-1245). | MR
[16] , Perturbation Theory for Linear Operators, Classics in Mathematics. Springer-Verlag, Berlin (1995). | MR | Zbl
[17] , Collected papers of L.D. Landau. Edited and with an introduction by D. ter Haar, Second edition, Gordon and Breach Science Publishers, New York (1967). | MR
[18] , On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator. Asymptot. Anal. 48 (2006) 295-357. | MR | Zbl
[19] , On the stability of a center for time-periodic perturbations. Trudy Moskov. Mat. Obšč. 12 (1963) 3-52. | MR | Zbl
[20] and , On the necessity of gaps. J. Eur. Math. Soc. (JEMS) 8 (2006) 355-373. | MR
[21] and , The geometry of mesoscopic phase transition interfaces. Discrete Contin. Dyn. Syst. 19 (2007) 777-798. | MR | Zbl
[22] , Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris (1892). | JFM
[23] and , Mixed states for an Allen-Cahn type equation. Comm. Pure Appl. Math. 56 (2003) 1078-1134. Dedicated to the memory of Jürgen K. Moser. | MR
[24] and , Mixed states for an Allen-Cahn type equation. II. Calc. Var. Partial Diff. Eq. 21 (2004) 157-207. | MR | Zbl
[25] , Translation of J.D. van der Waals' “The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density”. J. Statist. Phys. 20 (1979) 197-244. | MR
[26] , On the stability of the saddle solution of Allen-Cahn's equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1241-1275. | MR | Zbl
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