Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations
Matteo Novaga; Enrico Valdinoci
ESAIM: Control, Optimisation and Calculus of Variations (2009)
- Volume: 15, Issue: 4, page 914-933
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topNovaga, Matteo, and Valdinoci, Enrico. "Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2009): 914-933. <http://eudml.org/doc/245376>.
@article{Novaga2009,
abstract = {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.},
author = {Novaga, Matteo, Valdinoci, Enrico},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {oscillatory solutions of PDEs; phase transitions; asymptotic expansions; Ginzburg-Landau equation; Allen-Cahn equation; minimizer},
language = {eng},
number = {4},
pages = {914-933},
publisher = {EDP-Sciences},
title = {Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations},
url = {http://eudml.org/doc/245376},
volume = {15},
year = {2009},
}
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
PB - EDP-Sciences
VL - 15
IS - 4
SP - 914
EP - 933
AB - 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.
LA - eng
KW - oscillatory solutions of PDEs; phase transitions; asymptotic expansions; Ginzburg-Landau equation; Allen-Cahn equation; minimizer
UR - http://eudml.org/doc/245376
ER -
References
top- [1] F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered solutions in for a class of periodic Allen-Cahn equations. Comm. Partial Diff. Eq. 27 (2002) 1537–1574. Zbl1125.35342MR1924477
- [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. Zbl1004.37043MR1614571
- [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. Zbl1134.34337MR2278906
- [5] H. Brezis, Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983). Zbl0511.46001MR697382
- [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. Zbl0835.35045MR1340266
- [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. Zbl1152.35038MR2354993
- [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). Zbl1202.35313MR2515785
- [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. Zbl1101.35074MR2231253
- [10] N. Dirr, M. Lucia and M. Novaga, -convergence of the Allen-Cahn energy with an oscillating forcing term. Interfaces Free Bound. 8 (2006) 47–78. Zbl1106.49053MR2231252
- [11] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI (1998). Zbl0902.35002MR1625845
- [12] A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions. Calc. Var. Partial Differential Equations 33 (2008) 1–35. Zbl1156.35018MR2413100
- [13] G. Gallavotti, The elements of mechanics, Texts and Monographs in Physics. Springer-Verlag, New York (1983). Translated from the Italian. Zbl0512.70001MR698947
- [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). Zbl0562.35001MR737190
- [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). MR105929
- [16] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics. Springer-Verlag, Berlin (1995). Zbl0836.47009MR1335452
- [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). MR237287
- [18] M. Marx, On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator. Asymptot. Anal. 48 (2006) 295–357. Zbl1124.34063MR2256576
- [19] V.K. Mel’nikov, On the stability of a center for time-periodic perturbations. Trudy Moskov. Mat. Obšč. 12 (1963) 3–52. Zbl0135.31001MR156048
- [20] H. Matano and P.H. Rabinowitz, On the necessity of gaps. J. Eur. Math. Soc. (JEMS) 8 (2006) 355–373. Zbl1245.35043MR2239282
- [21] M. Novaga and E. Valdinoci, The geometry of mesoscopic phase transition interfaces. Discrete Contin. Dyn. Syst. 19 (2007) 777–798. Zbl1152.35005MR2342272
- [22] H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris (1892). JFM30.0834.08
- [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. Zbl1274.35122MR1989227
- [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. Zbl1161.35397MR2085301
- [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. Zbl1245.82006MR523642
- [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. Zbl0852.35020MR1363002
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.