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January  2013, 12(1): 303-340. doi: 10.3934/cpaa.2013.12.303

Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains

1. 

College of Mathematics and Computational Sciences, Shenzhen University, Nanhai Ave 3688, Shenzhen 518060

2. 

College of Mathematics and Computer Sciences, Hunan Normal University, Changsha 410081, China

Received  June 2011 Revised  December 2011 Published  September 2012

For a singularly perturbed equation of inhomogeneous Allen-Cahn type with positive potential function in high dimensional general domain, we prove the existence of solutions, at least for some sequence of the positive parameter, which have clustered phase transition layers with mass centered close to a smooth closed stationary and non-degenerate hypersurface. Moreover, the interaction between neighboring layers is governed by a type of Jacobi-Toda system.
Citation: Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure & Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303
References:
[1]

N. D. Alikakos and P. W. Bates, On the singular limit in a phase field model of phase transitions,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 5 (1988), 141.

[2]

N. D. Alikakos, P. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains,, Trans. Amer. Math. Soc., 351 (1999), 2777. doi: 10.1090/S0002-9947-99-02134-0.

[3]

N. D. Alikakos, P. W. Bates and G. Fusco, Solutions to the nonautonomous bistable equation with specified Morse index, I. Existence,, Trans. Amer. Math. Soc., 340 (1993), 641.

[4]

N. D. Alikakos, X. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray,, Calc. Var. Partial Differential Equations, 11 (2000), 233. doi: 10.1007/s005260000052.

[5]

N. D. Alikakos and H. C. Simpson, A variational approach for a class of singular perturbation problems and applications,, Proc. Roy. Soc. Edinburgh Sect. A, 107 (1987), 27. doi: 10.1017/S0308210500029334.

[6]

S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta. Metall., 27 (1979), 1085. doi: 10.1016/0001-6160(79)90196-2.

[7]

S. Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem,, J. Differential Equations, 67 (1987), 212. doi: 10.1016/0022-0396(87)90147-1.

[8]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces,, Math. Res. Lett., 3 (1996), 41.

[9]

I. Chavel, "Eigenvalues in Riemannian Geometry,'', Pure and Applied Mathematics, (1984).

[10]

I. Chavel, "Riemannian Geometry - A Modern Introduction,'', Cambridge Tracts in Math. 108, (1993).

[11]

E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem,, J. Differential Equations, 194 (2003), 382. doi: 10.1016/S0022-0396(03)00176-1.

[12]

E. N. Dancer and S. Yan, Construction of various types of solutions for an elliptic problem,, Calc. Var. Partial Differential Equations, 20 (2004), 93. doi: 10.1007/s00526-003-0229-6.

[13]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem,, Comm. Partial Differential Equations, 17 (1992), 1695. doi: 10.1080/03605309208820900.

[14]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system,, Trans. Amer. Math. Soc., 347 (1995), 4807. doi: 10.1090/S0002-9947-1995-1303116-3.

[15]

M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 60 (2007), 113. doi: 10.1002/cpa.20135.

[16]

M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model,, SIAM J. Math. Anal., 38 (): 1542. doi: 10.1137/060649574.

[17]

M. del Pino, M. Kowalczyk and J. Wei, The Toda system and clustering interface in the Allen-Cahn equation,, Arch. Ration. Mech. Anal., 190 (2008), 141. doi: 10.1007/s00205-008-0143-3.

[18]

M. del Pino, M. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces,, Discrete Contin. Dynam. Systems-A, 28 (2010), 975. doi: 10.3934/dcds.2010.28.975.

[19]

M. del Pino, M. Kowalczyk, J. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature,, Geom. Funct. Anal., 20 (2010), 918. doi: 10.1007/s00039-010-0083-6.

[20]

A. S. do Nascimento, Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in $N$-dimensional domains,, J. Differential Equations, 190 (2003), 16. doi: 10.1016/S0022-0396(02)00147-X.

[21]

Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation,, J. Differential Equations, 249 (2010), 215. doi: 10.1016/j.jde.2010.03.024.

[22]

P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations,, J. Math. Anal. Appl., 54 (1976), 497. doi: 10.1016/0022-247X(76)90218-3.

[23]

P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter,, Russian Math. Survey, 29 (1974), 103. doi: 10.1070/RM1974v029n04ABEH001291.

[24]

G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach,, J. Differential Equations, 169 (2001), 190. doi: 10.1006/jdeq.2000.3898.

[25]

C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation,, Comm. Anal. Geom., 11 (2003), 223.

[26]

J. Hale and K. Sakamoto, Existence and stability of transition layers,, Japan J. Appl. Math., 5 (1988), 367. doi: 10.1007/BF03167908.

[27]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations,, Proc. Royal Soc. Edinburgh A, 111 (1989), 69. doi: 10.1017/S0308210500025026.

[28]

M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions,, Annali di Matematica Pura ed Applicata, 184 (2005), 17. doi: 10.1007/s10231-003-0088-y.

[29]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains,, Discrete Contin. Dynam. Systems-A, 32 (2012), 1391. doi: 10.3934/dcds.2012.32.1391.

[30]

P. Li and S. Yau, On the Schrödinger equation and the eigenvalue problem,, Commun. Math. Phys., 88 (1983), 309. doi: 10.1007/BF01213210.

[31]

F. Mahmoudi, A. Malchiodi and J. Wei, Transition layer for the heterogeneous Allen-Cahn equation,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 25 (2008), 609. doi: 10.1016/j.anihpc.2007.03.008.

[32]

A. Malchiodi, W.-M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation,, Pacific J. Math., 229 (2007), 447. doi: 10.2140/pjm.2007.229.447.

[33]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation,, Journal of Fixed Point Theory and Applications, 1 (2007), 305. doi: 10.1007/s11784-007-0016-7.

[34]

S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds,, Canad. J. Math., 1 (1949), 242. doi: 10.4153/CJM-1949-021-5.

[35]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rat. Mech. Anal., 98 (1987), 123. doi: 10.1007/BF00251230.

[36]

K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation,, J. Differential Equations, 191 (2003), 234. doi: 10.1016/S0022-0396(02)00181-X.

[37]

K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 20 (2003), 107. doi: 10.1016/S0294-1449(02)00008-2.

[38]

Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. doi: 10.1137/0518124.

[39]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions,, J. Diff. Geom., 64 (2003), 359.

[40]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions,, Comm. Pure Appl. Math., 51 (1998), 551. doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.

[41]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I,, Comm Pure Appl. Math., 56 (2003), 1078. doi: 10.1002/cpa.10087.

[42]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II,, Calc. Var. Partial Differential Equations, 21 (2004), 157. doi: 10.1007/s00526-003-0251-8.

[43]

K. Sakamoto, Construction and stability an alysis of transition layer solutions in reaction-diffusion systems,, Tohoku Math. J., 42 (1990), 17. doi: 10.2748/tmj/1178227692.

[44]

K. Sakamoto, Infinitely many fine modes bifurcating from radially symmetric internal layers,, Asymptot. Anal., 42 (2005), 55.

[45]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081.

[46]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model,, Asymptot. Anal., 69 (2010), 175.

show all references

References:
[1]

N. D. Alikakos and P. W. Bates, On the singular limit in a phase field model of phase transitions,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 5 (1988), 141.

[2]

N. D. Alikakos, P. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains,, Trans. Amer. Math. Soc., 351 (1999), 2777. doi: 10.1090/S0002-9947-99-02134-0.

[3]

N. D. Alikakos, P. W. Bates and G. Fusco, Solutions to the nonautonomous bistable equation with specified Morse index, I. Existence,, Trans. Amer. Math. Soc., 340 (1993), 641.

[4]

N. D. Alikakos, X. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray,, Calc. Var. Partial Differential Equations, 11 (2000), 233. doi: 10.1007/s005260000052.

[5]

N. D. Alikakos and H. C. Simpson, A variational approach for a class of singular perturbation problems and applications,, Proc. Roy. Soc. Edinburgh Sect. A, 107 (1987), 27. doi: 10.1017/S0308210500029334.

[6]

S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta. Metall., 27 (1979), 1085. doi: 10.1016/0001-6160(79)90196-2.

[7]

S. Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem,, J. Differential Equations, 67 (1987), 212. doi: 10.1016/0022-0396(87)90147-1.

[8]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces,, Math. Res. Lett., 3 (1996), 41.

[9]

I. Chavel, "Eigenvalues in Riemannian Geometry,'', Pure and Applied Mathematics, (1984).

[10]

I. Chavel, "Riemannian Geometry - A Modern Introduction,'', Cambridge Tracts in Math. 108, (1993).

[11]

E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem,, J. Differential Equations, 194 (2003), 382. doi: 10.1016/S0022-0396(03)00176-1.

[12]

E. N. Dancer and S. Yan, Construction of various types of solutions for an elliptic problem,, Calc. Var. Partial Differential Equations, 20 (2004), 93. doi: 10.1007/s00526-003-0229-6.

[13]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem,, Comm. Partial Differential Equations, 17 (1992), 1695. doi: 10.1080/03605309208820900.

[14]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system,, Trans. Amer. Math. Soc., 347 (1995), 4807. doi: 10.1090/S0002-9947-1995-1303116-3.

[15]

M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 60 (2007), 113. doi: 10.1002/cpa.20135.

[16]

M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model,, SIAM J. Math. Anal., 38 (): 1542. doi: 10.1137/060649574.

[17]

M. del Pino, M. Kowalczyk and J. Wei, The Toda system and clustering interface in the Allen-Cahn equation,, Arch. Ration. Mech. Anal., 190 (2008), 141. doi: 10.1007/s00205-008-0143-3.

[18]

M. del Pino, M. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces,, Discrete Contin. Dynam. Systems-A, 28 (2010), 975. doi: 10.3934/dcds.2010.28.975.

[19]

M. del Pino, M. Kowalczyk, J. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature,, Geom. Funct. Anal., 20 (2010), 918. doi: 10.1007/s00039-010-0083-6.

[20]

A. S. do Nascimento, Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in $N$-dimensional domains,, J. Differential Equations, 190 (2003), 16. doi: 10.1016/S0022-0396(02)00147-X.

[21]

Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation,, J. Differential Equations, 249 (2010), 215. doi: 10.1016/j.jde.2010.03.024.

[22]

P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations,, J. Math. Anal. Appl., 54 (1976), 497. doi: 10.1016/0022-247X(76)90218-3.

[23]

P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter,, Russian Math. Survey, 29 (1974), 103. doi: 10.1070/RM1974v029n04ABEH001291.

[24]

G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach,, J. Differential Equations, 169 (2001), 190. doi: 10.1006/jdeq.2000.3898.

[25]

C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation,, Comm. Anal. Geom., 11 (2003), 223.

[26]

J. Hale and K. Sakamoto, Existence and stability of transition layers,, Japan J. Appl. Math., 5 (1988), 367. doi: 10.1007/BF03167908.

[27]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations,, Proc. Royal Soc. Edinburgh A, 111 (1989), 69. doi: 10.1017/S0308210500025026.

[28]

M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions,, Annali di Matematica Pura ed Applicata, 184 (2005), 17. doi: 10.1007/s10231-003-0088-y.

[29]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains,, Discrete Contin. Dynam. Systems-A, 32 (2012), 1391. doi: 10.3934/dcds.2012.32.1391.

[30]

P. Li and S. Yau, On the Schrödinger equation and the eigenvalue problem,, Commun. Math. Phys., 88 (1983), 309. doi: 10.1007/BF01213210.

[31]

F. Mahmoudi, A. Malchiodi and J. Wei, Transition layer for the heterogeneous Allen-Cahn equation,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 25 (2008), 609. doi: 10.1016/j.anihpc.2007.03.008.

[32]

A. Malchiodi, W.-M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation,, Pacific J. Math., 229 (2007), 447. doi: 10.2140/pjm.2007.229.447.

[33]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation,, Journal of Fixed Point Theory and Applications, 1 (2007), 305. doi: 10.1007/s11784-007-0016-7.

[34]

S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds,, Canad. J. Math., 1 (1949), 242. doi: 10.4153/CJM-1949-021-5.

[35]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rat. Mech. Anal., 98 (1987), 123. doi: 10.1007/BF00251230.

[36]

K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation,, J. Differential Equations, 191 (2003), 234. doi: 10.1016/S0022-0396(02)00181-X.

[37]

K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 20 (2003), 107. doi: 10.1016/S0294-1449(02)00008-2.

[38]

Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. doi: 10.1137/0518124.

[39]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions,, J. Diff. Geom., 64 (2003), 359.

[40]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions,, Comm. Pure Appl. Math., 51 (1998), 551. doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.

[41]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I,, Comm Pure Appl. Math., 56 (2003), 1078. doi: 10.1002/cpa.10087.

[42]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II,, Calc. Var. Partial Differential Equations, 21 (2004), 157. doi: 10.1007/s00526-003-0251-8.

[43]

K. Sakamoto, Construction and stability an alysis of transition layer solutions in reaction-diffusion systems,, Tohoku Math. J., 42 (1990), 17. doi: 10.2748/tmj/1178227692.

[44]

K. Sakamoto, Infinitely many fine modes bifurcating from radially symmetric internal layers,, Asymptot. Anal., 42 (2005), 55.

[45]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081.

[46]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model,, Asymptot. Anal., 69 (2010), 175.

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