×

Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations. (English) Zbl 1196.35202

One considers the functional \[ \int_{\mathbb R^n} \frac{{|\nabla u|}^2}{2} + F(u) + H(x)u dx \] whose Euler-Lagrange equation is an Allen-Cahn type equation which includes the well-known Ginzburg-Landau equation \((H(x) = 0)\). In their earlier paper [Discrete Contin. Dyn. Syst. 19, No. 4, 777–798 (2007; Zbl 1152.35005)] the authors proved the existence of two periodic minimizers of the functional. Based on these results multibump solutions of the mesoscopic model for phase transitions are constructed in the present publication using the mesoscopic term \(H(x)\) and eigenvalue relations.

MSC:

35Q56 Ginzburg-Landau equations
35B40 Asymptotic behavior of solutions to PDEs
49R05 Variational methods for eigenvalues of operators
35C20 Asymptotic expansions of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Citations:

Zbl 1152.35005
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered solutions in \(\mathbb{R}^2\) for a class of periodic Allen-Cahn equations. Comm. Partial Diff. Eq. 27 (2002) 1537-1574. Zbl1125.35342 MR1924477 · Zbl 1125.35342 · doi:10.1081/PDE-120005848
[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. Zbl1004.37043 MR1614571 · Zbl 1004.37043 · doi:10.1016/S0294-1449(97)89300-6
[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. Zbl1134.34337 MR2278906 · Zbl 1134.34337
[5] H. Brezis, Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983). Zbl0511.46001 MR697382 · Zbl 0511.46001
[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. Zbl0835.35045 MR1340266 · Zbl 0835.35045
[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. Zbl1152.35038 MR2354993 · Zbl 1152.35038 · doi:10.1016/j.aim.2007.03.013
[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). MR2515785 · Zbl 1202.35313
[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. Zbl1101.35074 MR2231253 · Zbl 1101.35074 · doi:10.4171/IFB/136
[10] N. Dirr, M. Lucia and M. Novaga, \(\Gamma \)-convergence of the Allen-Cahn energy with an oscillating forcing term. Interfaces Free Bound. 8 (2006) 47-78. Zbl1106.49053 MR2231252 · Zbl 1106.49053 · doi:10.4171/IFB/135
[11] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI (1998). Zbl0902.35002 MR1625845 · Zbl 0902.35002
[12] A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions. Calc. Var. Partial Differential Equations 33 (2008) 1-35. Zbl1156.35018 MR2413100 · Zbl 1156.35018 · doi:10.1007/s00526-007-0146-1
[13] G. Gallavotti, The elements of mechanics, Texts and Monographs in Physics. Springer-Verlag, New York (1983). Translated from the Italian. Zbl0512.70001 MR698947 · Zbl 0512.70001
[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). Zbl0562.35001 MR737190 · Zbl 0562.35001
[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). Zbl0836.47009 MR1335452 · Zbl 0836.47009
[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. Zbl1124.34063 MR2256576 · Zbl 1124.34063
[19] V.K. Mel’nikov, On the stability of a center for time-periodic perturbations. Trudy Moskov. Mat. Obšč. 12 (1963) 3-52. Zbl0135.31001 MR156048 · Zbl 0135.31001
[20] H. Matano and P.H. Rabinowitz, On the necessity of gaps. J. Eur. Math. Soc. (JEMS) 8 (2006) 355-373. Zblpre05053369 MR2239282 · Zbl 1245.35043
[21] M. Novaga and E. Valdinoci, The geometry of mesoscopic phase transition interfaces. Discrete Contin. Dyn. Syst. 19 (2007) 777-798. Zbl1152.35005 MR2342272 · Zbl 1152.35005 · doi:10.3934/dcds.2007.19.777
[22] H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris (1892). JFM30.0834.08 · JFM 30.0834.08
[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. Zblpre01981618 MR1989227 · Zbl 1274.35122
[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. Zbl1161.35397 MR2085301 · Zbl 1161.35397 · doi:10.1007/s00526-003-0251-8
[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. MR523642 · Zbl 1245.82006
[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. Zbl0852.35020 MR1363002 · Zbl 0852.35020 · doi:10.1017/S0308210500030493
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.