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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

Jun Yang 1, and  Xiaolin Yang 2,

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.
Keywords: inhomogeneous Allen-Cahn equations, Multiple phase transition layers, Jacobi-Toda system..
Mathematics Subject Classification: Primary: 35J60; Secondary: 58J2.
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. Poincaré Anal. Non Linéaire, 5 (1988), 141-178.  Google Scholar

[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), 2777-2805. doi: 10.1090/S0002-9947-99-02134-0.  Google Scholar

[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), 641-654.  Google Scholar

[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), 233-305. doi: 10.1007/s005260000052.  Google Scholar

[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), 27-42. doi: 10.1017/S0308210500029334.  Google Scholar

[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), 1085-1095. doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[7]

S. Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem, J. Differential Equations, 67 (1987), 212-242. doi: 10.1016/0022-0396(87)90147-1.  Google Scholar

[8]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 41-50.  Google Scholar

[9]

I. Chavel, "Eigenvalues in Riemannian Geometry,'' Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984.  Google Scholar

[10]

I. Chavel, "Riemannian Geometry - A Modern Introduction,'' Cambridge Tracts in Math. 108, Cambridge Univ. Press, Cambridge, 1993.  Google Scholar

[11]

E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem, J. Differential Equations, 194 (2003), 382-405. doi: 10.1016/S0022-0396(03)00176-1.  Google Scholar

[12]

E. N. Dancer and S. Yan, Construction of various types of solutions for an elliptic problem, Calc. Var. Partial Differential Equations, 20 (2004), 93-118. doi: 10.1007/s00526-003-0229-6.  Google Scholar

[13]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem, Comm. Partial Differential Equations, 17 (1992), 1695-1708. doi: 10.1080/03605309208820900.  Google Scholar

[14]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837. doi: 10.1090/S0002-9947-1995-1303116-3.  Google Scholar

[15]

M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.  Google Scholar

[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.  Google Scholar

[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), 141-187. doi: 10.1007/s00205-008-0143-3.  Google Scholar

[18]

M. del Pino, M. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dynam. Systems-A, 28 (2010), 975-1006. doi: 10.3934/dcds.2010.28.975.  Google Scholar

[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), 918-957. doi: 10.1007/s00039-010-0083-6.  Google Scholar

[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), 16-38. doi: 10.1016/S0022-0396(02)00147-X.  Google Scholar

[21]

Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Differential Equations, 249 (2010), 215-239. doi: 10.1016/j.jde.2010.03.024.  Google Scholar

[22]

P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521. doi: 10.1016/0022-247X(76)90218-3.  Google Scholar

[23]

P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter, Russian Math. Survey, 29 (1974), 103-131. doi: 10.1070/RM1974v029n04ABEH001291.  Google Scholar

[24]

G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach, J. Differential Equations, 169 (2001), 190-207. doi: 10.1006/jdeq.2000.3898.  Google Scholar

[25]

C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation, Comm. Anal. Geom., 11 (2003), 223-233.  Google Scholar

[26]

J. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405. doi: 10.1007/BF03167908.  Google Scholar

[27]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh A, 111 (1989), 69-84. doi: 10.1017/S0308210500025026.  Google Scholar

[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), 17-52. doi: 10.1007/s10231-003-0088-y.  Google Scholar

[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), 1391-1420. doi: 10.3934/dcds.2012.32.1391.  Google Scholar

[30]

P. Li and S. Yau, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88 (1983), 309-318. doi: 10.1007/BF01213210.  Google Scholar

[31]

F. Mahmoudi, A. Malchiodi and J. Wei, Transition layer for the heterogeneous Allen-Cahn equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 609-631. doi: 10.1016/j.anihpc.2007.03.008.  Google Scholar

[32]

A. Malchiodi, W.-M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation, Pacific J. Math., 229 (2007), 447-468. doi: 10.2140/pjm.2007.229.447.  Google Scholar

[33]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation, Journal of Fixed Point Theory and Applications, 1 (2007), 305-336. doi: 10.1007/s11784-007-0016-7.  Google Scholar

[34]

S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math., 1 (1949), 242-256. doi: 10.4153/CJM-1949-021-5.  Google Scholar

[35]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.  Google Scholar

[36]

K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Differential Equations, 191 (2003), 234-276. doi: 10.1016/S0022-0396(02)00181-X.  Google Scholar

[37]

K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 107-143. doi: 10.1016/S0294-1449(02)00008-2.  Google Scholar

[38]

Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770. doi: 10.1137/0518124.  Google Scholar

[39]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Diff. Geom., 64 (2003), 359-423.  Google Scholar

[40]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Comm. Pure Appl. Math., 51 (1998), 551-579. doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.  Google Scholar

[41]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I, Comm Pure Appl. Math., 56 (2003), 1078-1134. doi: 10.1002/cpa.10087.  Google Scholar

[42]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II, Calc. Var. Partial Differential Equations, 21 (2004), 157-207. doi: 10.1007/s00526-003-0251-8.  Google Scholar

[43]

K. Sakamoto, Construction and stability an alysis of transition layer solutions in reaction-diffusion systems, Tohoku Math. J., 42 (1990), 17-44. doi: 10.2748/tmj/1178227692.  Google Scholar

[44]

K. Sakamoto, Infinitely many fine modes bifurcating from radially symmetric internal layers, Asymptot. Anal., 42 (2005), 55-104.  Google Scholar

[45]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400. doi: 10.1007/s002050050081.  Google Scholar

[46]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model, Asymptot. Anal., 69 (2010), 175-218.  Google Scholar

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. Poincaré Anal. Non Linéaire, 5 (1988), 141-178.  Google Scholar

[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), 2777-2805. doi: 10.1090/S0002-9947-99-02134-0.  Google Scholar

[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), 641-654.  Google Scholar

[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), 233-305. doi: 10.1007/s005260000052.  Google Scholar

[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), 27-42. doi: 10.1017/S0308210500029334.  Google Scholar

[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), 1085-1095. doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[7]

S. Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem, J. Differential Equations, 67 (1987), 212-242. doi: 10.1016/0022-0396(87)90147-1.  Google Scholar

[8]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 41-50.  Google Scholar

[9]

I. Chavel, "Eigenvalues in Riemannian Geometry,'' Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984.  Google Scholar

[10]

I. Chavel, "Riemannian Geometry - A Modern Introduction,'' Cambridge Tracts in Math. 108, Cambridge Univ. Press, Cambridge, 1993.  Google Scholar

[11]

E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem, J. Differential Equations, 194 (2003), 382-405. doi: 10.1016/S0022-0396(03)00176-1.  Google Scholar

[12]

E. N. Dancer and S. Yan, Construction of various types of solutions for an elliptic problem, Calc. Var. Partial Differential Equations, 20 (2004), 93-118. doi: 10.1007/s00526-003-0229-6.  Google Scholar

[13]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem, Comm. Partial Differential Equations, 17 (1992), 1695-1708. doi: 10.1080/03605309208820900.  Google Scholar

[14]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837. doi: 10.1090/S0002-9947-1995-1303116-3.  Google Scholar

[15]

M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.  Google Scholar

[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.  Google Scholar

[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), 141-187. doi: 10.1007/s00205-008-0143-3.  Google Scholar

[18]

M. del Pino, M. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dynam. Systems-A, 28 (2010), 975-1006. doi: 10.3934/dcds.2010.28.975.  Google Scholar

[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), 918-957. doi: 10.1007/s00039-010-0083-6.  Google Scholar

[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), 16-38. doi: 10.1016/S0022-0396(02)00147-X.  Google Scholar

[21]

Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Differential Equations, 249 (2010), 215-239. doi: 10.1016/j.jde.2010.03.024.  Google Scholar

[22]

P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521. doi: 10.1016/0022-247X(76)90218-3.  Google Scholar

[23]

P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter, Russian Math. Survey, 29 (1974), 103-131. doi: 10.1070/RM1974v029n04ABEH001291.  Google Scholar

[24]

G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach, J. Differential Equations, 169 (2001), 190-207. doi: 10.1006/jdeq.2000.3898.  Google Scholar

[25]

C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation, Comm. Anal. Geom., 11 (2003), 223-233.  Google Scholar

[26]

J. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405. doi: 10.1007/BF03167908.  Google Scholar

[27]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh A, 111 (1989), 69-84. doi: 10.1017/S0308210500025026.  Google Scholar

[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), 17-52. doi: 10.1007/s10231-003-0088-y.  Google Scholar

[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), 1391-1420. doi: 10.3934/dcds.2012.32.1391.  Google Scholar

[30]

P. Li and S. Yau, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88 (1983), 309-318. doi: 10.1007/BF01213210.  Google Scholar

[31]

F. Mahmoudi, A. Malchiodi and J. Wei, Transition layer for the heterogeneous Allen-Cahn equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 609-631. doi: 10.1016/j.anihpc.2007.03.008.  Google Scholar

[32]

A. Malchiodi, W.-M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation, Pacific J. Math., 229 (2007), 447-468. doi: 10.2140/pjm.2007.229.447.  Google Scholar

[33]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation, Journal of Fixed Point Theory and Applications, 1 (2007), 305-336. doi: 10.1007/s11784-007-0016-7.  Google Scholar

[34]

S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math., 1 (1949), 242-256. doi: 10.4153/CJM-1949-021-5.  Google Scholar

[35]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.  Google Scholar

[36]

K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Differential Equations, 191 (2003), 234-276. doi: 10.1016/S0022-0396(02)00181-X.  Google Scholar

[37]

K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 107-143. doi: 10.1016/S0294-1449(02)00008-2.  Google Scholar

[38]

Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770. doi: 10.1137/0518124.  Google Scholar

[39]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Diff. Geom., 64 (2003), 359-423.  Google Scholar

[40]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Comm. Pure Appl. Math., 51 (1998), 551-579. doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.  Google Scholar

[41]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I, Comm Pure Appl. Math., 56 (2003), 1078-1134. doi: 10.1002/cpa.10087.  Google Scholar

[42]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II, Calc. Var. Partial Differential Equations, 21 (2004), 157-207. doi: 10.1007/s00526-003-0251-8.  Google Scholar

[43]

K. Sakamoto, Construction and stability an alysis of transition layer solutions in reaction-diffusion systems, Tohoku Math. J., 42 (1990), 17-44. doi: 10.2748/tmj/1178227692.  Google Scholar

[44]

K. Sakamoto, Infinitely many fine modes bifurcating from radially symmetric internal layers, Asymptot. Anal., 42 (2005), 55-104.  Google Scholar

[45]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400. doi: 10.1007/s002050050081.  Google Scholar

[46]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model, Asymptot. Anal., 69 (2010), 175-218.  Google Scholar

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