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.

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.

DOI : 10.1051/cocv:2008058
Classification : 35B40, 49R50
Mots clés : oscillatory solutions of PDEs, phase transitions, asymptotic expansions
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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://archive.numdam.org/articles/10.1051/cocv:2008058/

[1] F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered solutions in 2 for a class of periodic Allen-Cahn equations. Comm. Partial Diff. Eq. 27 (2002) 1537-1574. | MR | Zbl

[2] S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1084-1095.

[3] A. Ambrosetti and M. Badiale, Homoclinics: Poincaré-Melnikov type results via a variational approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 233-252. | Numdam | MR | Zbl

[4] D.I. Borisov, 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] H. Brezis, Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983). | MR | Zbl

[6] G. Carbou, 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] R. De La Llave and E. Valdinoci, Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media. Adv. Math. 215 (2007) 379-426. | MR | Zbl

[8] N. Dirr and E. Orlandi, 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] N. Dirr and N.K. Yip, Pinning and de-pinning phenomena in front propagation in heterogeneous media. Interfaces Free Bound. 8 (2006) 79-109. | MR | Zbl

[10] N. Dirr, M. Lucia and M. Novaga, Γ-convergence of the Allen-Cahn energy with an oscillating forcing term. Interfaces Free Bound. 8 (2006) 47-78. | MR | Zbl

[11] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI (1998). | MR | Zbl

[12] A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions. Calc. Var. Partial Differential Equations 33 (2008) 1-35. | MR | Zbl

[13] G. Gallavotti, The elements of mechanics, Texts and Monographs in Physics. Springer-Verlag, New York (1983). Translated from the Italian. | MR | Zbl

[14] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften 224. Springer-Verlag, Berlin, second edition (1983). | MR | Zbl

[15] V.L. Ginzburg and L.P. Pitaevskiĭ, On the theory of superfluidity. Soviet Physics. JETP 34 (1958) 858-861 (Ž. Eksper. Teoret. Fiz. 1240-1245). | MR

[16] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics. Springer-Verlag, Berlin (1995). | MR | Zbl

[17] L.D. Landau, 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] M. Marx, On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator. Asymptot. Anal. 48 (2006) 295-357. | MR | Zbl

[19] V.K. Mel'Nikov, On the stability of a center for time-periodic perturbations. Trudy Moskov. Mat. Obšč. 12 (1963) 3-52. | MR | Zbl

[20] H. Matano and P.H. Rabinowitz, On the necessity of gaps. J. Eur. Math. Soc. (JEMS) 8 (2006) 355-373. | MR

[21] M. Novaga and E. Valdinoci, The geometry of mesoscopic phase transition interfaces. Discrete Contin. Dyn. Syst. 19 (2007) 777-798. | MR | Zbl

[22] H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris (1892). | JFM

[23] P.H. Rabinowitz and E. Stredulinsky, 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] P.H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation. II. Calc. Var. Partial Diff. Eq. 21 (2004) 157-207. | MR | Zbl

[25] J.S. Rowlinson, 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] M. Schatzman, 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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