Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations

Matteo Novaga; Enrico Valdinoci

doi:10.1051/cocv:2008058

All issues

Volume 15 / No 4 (October-December 2009)

ESAIM: COCV, 15 4 (2009) 914-933

References

Free Access

Issue		ESAIM: COCV Volume 15, Number 4, October-December 2009


Page(s)		914 - 933
DOI		https://doi.org/10.1051/cocv:2008058
Published online		20 August 2008

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. [CrossRef]
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.
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. [CrossRef] [MathSciNet]
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. [CrossRef] [MathSciNet]
H. Brezis, Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983).
G. Carbou, Unicité et minimalité des solutions d'une équation de Ginzburg-Landau. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 305–318.
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. [CrossRef] [MathSciNet]
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).
N. Dirr and N.K. Yip, Pinning and de-pinning phenomena in front propagation in heterogeneous media. Interfaces Free Bound. 8 (2006) 79–109. [CrossRef] [MathSciNet]
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. [CrossRef] [MathSciNet]
L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI (1998).
A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions. Calc. Var. Partial Differential Equations 33 (2008) 1–35. [CrossRef] [MathSciNet]
G. Gallavotti, The elements of mechanics, Texts and Monographs in Physics. Springer-Verlag, New York (1983). Translated from the Italian.
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften 224. Springer-Verlag, Berlin, second edition (1983).
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). [MathSciNet]
T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics. Springer-Verlag, Berlin (1995).
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).
M. Marx, On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator. Asymptot. Anal. 48 (2006) 295–357. [MathSciNet]
V.K. Mel'nikov, On the stability of a center for time-periodic perturbations. Trudy Moskov. Mat. Obšč. 12 (1963) 3–52. [MathSciNet]
H. Matano and P.H. Rabinowitz, On the necessity of gaps. J. Eur. Math. Soc. (JEMS) 8 (2006) 355–373. [CrossRef] [MathSciNet]
M. Novaga and E. Valdinoci, The geometry of mesoscopic phase transition interfaces. Discrete Contin. Dyn. Syst. 19 (2007) 777–798. [CrossRef] [MathSciNet]
H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris (1892).
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. [CrossRef] [MathSciNet]
P.H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation. II. Calc. Var. Partial Diff. Eq. 21 (2004) 157–207.
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. [CrossRef] [MathSciNet]
M. Schatzman, On the stability of the saddle solution of Allen-Cahn's equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1241–1275. [MathSciNet]

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[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. [CrossRef]

[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. [CrossRef] [MathSciNet]

[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. [CrossRef] [MathSciNet]

[5] H. Brezis, Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983).

[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.

[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. [CrossRef] [MathSciNet]

[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).

[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. [CrossRef] [MathSciNet]

[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. [CrossRef] [MathSciNet]

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

[12] A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions. Calc. Var. Partial Differential Equations 33 (2008) 1–35. [CrossRef] [MathSciNet]

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

[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).

[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). [MathSciNet]

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

[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).

[18] M. Marx, On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator. Asymptot. Anal. 48 (2006) 295–357. [MathSciNet]

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

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

[21] M. Novaga and E. Valdinoci, The geometry of mesoscopic phase transition interfaces. Discrete Contin. Dyn. Syst. 19 (2007) 777–798. [CrossRef] [MathSciNet]

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

[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. [CrossRef] [MathSciNet]

[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.

[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. [CrossRef] [MathSciNet]

[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. [MathSciNet]