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May  2020, 19(5): 2575-2616. doi: 10.3934/cpaa.2020113

Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation

1. 

Department of Mathematics, South China Agricultural University, Guangzhou 510642, China,

2. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

3. 

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

* Corresponding author

Received  December 2018 Revised  November 2019 Published  March 2020

Fund Project: Jun Yang is supported by National Natural Science Foundation of China (Grant No. 11771167 and No. 11831009)

We consider the nonlinear problem of inhomogeneous Allen-Cahn equation
$ \epsilon^2\Delta u+V(y)\,(1-u^2)\,u = 0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu} = 0\quad \mbox{on}\ \partial \Omega, $
where
$ \Omega $
is a bounded domain in
$ \mathbb R^2 $
with smooth boundary,
$ \epsilon $
is a small positive parameter,
$ \nu $
denotes the unit outward normal of
$ \partial \Omega $
,
$ V $
is a positive smooth function on
$ \bar\Omega $
. Let
$ \Gamma\subset\Omega $
be a smooth curve dividing
$ \Omega $
into two disjoint regions and intersecting orthogonally with
$ \partial\Omega $
at exactly two points
$ P_1 $
and
$ P_2 $
. Moreover, by considering
$ {\mathbb R}^2 $
as a Riemannian manifold with the metric
$ g = V(y)\,({\mathrm d}{y}_1^2+{\mathrm d}{y}_2^2) $
, we assume that: the curve
$ \Gamma $
is a non-degenerate geodesic in the Riemannian manifold
$ ({\mathbb R}^2, g) $
, the Ricci curvature of the Riemannian manifold
$ ({\mathbb R}^2, g) $
along the normal
$ \mathbf{n} $
of
$ \Gamma $
is positive at
$ \Gamma $
, the generalized mean curvature of the submanifold
$ \partial\Omega $
in
$ ({\mathbb R}^2, g) $
vanishes at
$ P_1 $
and
$ P_2 $
. Then for any given integer
$ N\geq 2 $
, we construct a solution exhibiting
$ N $
-phase transition layers near
$ \Gamma $
(the zero set of the solution has
$ N $
components, which are curves connecting
$ \partial\Omega $
and directed along the direction of
$ \Gamma $
) with mutual distance
$ O(\epsilon|\log \epsilon|) $
, provided that
$ \epsilon $
stays away from a discrete set of values to avoid the resonance of the problem. Asymptotic locations of these layers are governed by a Toda system.
Citation: Suting Wei, Jun Yang. Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2575-2616. doi: 10.3934/cpaa.2020113
References:
[1]

N. D. AlikakosP. 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.  doi: 10.2307/2154670.

[2]

N. D. AlikakosX. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray, Calc. Var. Partial Differ. Equ., 11 (2000), 233-305.  doi: 10.1007/s005260000052.

[3]

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.

[4]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 41-50.  doi: 10.4310/MRL.1996.v3.n1.a4.

[5]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem, Commun. Partial Differ. Equ., 17 (1992), 1695-1708.  doi: 10.1080/03605309208820900.

[6]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837.  doi: 10.2307/2155064.

[7]

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

[8]

M. del PinoM. 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.

[9]

M. del PinoM. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dyn. Syst., 28 (2010), 975-1006.  doi: 10.3934/dcds.2010.28.975.

[10]

M. del PinoM. KowalczykJ. 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.

[11]

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1976.

[12]

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

[13]

Z. Du and L. Wang, Interface foliation for an inhomogeneous Allen-Cahn equation in Riemannian manifolds, Calc. Var. Partial Differ. Equ., 47 (2013), 343-381.  doi: 10.1007/s00526-012-0521-4.

[14]

X. FanB. Xu and J. Yang, Phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation, J. Differ. Equ., 266 (2019), 5821-5866.  doi: 10.1016/j.jde.2018.10.051.

[15]

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

[16]

C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation, Comm. Anal. Geom., 11 (2003), 223-233.  doi: 10.4310/CAG.2003.v11.n2.a3.

[17]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. R. Soc. Edinb. Sect. A Math., 111 (1989), 69-84.  doi: 10.1017/S0308210500025026.

[18]

M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions, Ann. Mat. Pura Appl., 184 (2005), 17-52.  doi: 10.1007/s10231-003-0088-y.

[19]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains, Discrete Contin. Dyn. Syst.-A, 32 (2012), 1391-1420.  doi: 10.3934/dcds.2012.32.1391.

[20]

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

[21]

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

[22]

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

[23]

F. Morgan, Manifolds with Density, Notices Amer. Math. Soc., 52 (2005), 853-858. 

[24]

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

[25]

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

[26]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differ. Geom., 64 (2003), 359-423.  doi: 10.4310/jdg/1090426999.

[27]

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

[28]

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

[29]

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

[30]

K. Sakamoto, Existence and stability of three-dimensional boundary-interior layers for the Allen-Cahn equation, Taiwan. J. Math., 9 (2005), 331-358.  doi: 10.11650/twjm/1500407844.

[31]

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

[32]

F. TangS. Wei and J. Yang, Phase transition layers for Fife-Greenlee problem on smooth bounded domain, Discrete Contin. Dyn. Syst.-A, 38 (2018), 1527-1552.  doi: 10.3934/dcds.2018063.

[33]

J. Wei and J. Yang, Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain, Discrete Contin. Dyn. Syst.-A, 22 (2008), 465-508. doi: 10.3934/dcds.2008.22.465.

[34]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model, Asymptotic Anal., 69 (2010), 175-218.  doi: 10.3233/ASY-2010-0999.

[35]

S. Wei, B. Xu and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains, Calc. Var. Partial Differ. Equ., 57 (2018), Article: 87. doi: 10.1007/s00526-018-1347-5.

[36]

S. Wei and J. Yang, Connectivity of boundaries by clustering phase transition layers of Fife-Greenlee problem on smooth bounded domain, J. Differ. Equ., to appear. doi: 10.1016/j.jde.2020.01.014.

[37]

J. Yang and X. Yang, Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains, Commun. Pure Appl. Anal., 12 (2013), 303-340.  doi: 10.3934/cpaa.2013.12.303.

show all references

References:
[1]

N. D. AlikakosP. 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.  doi: 10.2307/2154670.

[2]

N. D. AlikakosX. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray, Calc. Var. Partial Differ. Equ., 11 (2000), 233-305.  doi: 10.1007/s005260000052.

[3]

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.

[4]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 41-50.  doi: 10.4310/MRL.1996.v3.n1.a4.

[5]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem, Commun. Partial Differ. Equ., 17 (1992), 1695-1708.  doi: 10.1080/03605309208820900.

[6]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837.  doi: 10.2307/2155064.

[7]

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

[8]

M. del PinoM. 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.

[9]

M. del PinoM. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dyn. Syst., 28 (2010), 975-1006.  doi: 10.3934/dcds.2010.28.975.

[10]

M. del PinoM. KowalczykJ. 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.

[11]

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1976.

[12]

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

[13]

Z. Du and L. Wang, Interface foliation for an inhomogeneous Allen-Cahn equation in Riemannian manifolds, Calc. Var. Partial Differ. Equ., 47 (2013), 343-381.  doi: 10.1007/s00526-012-0521-4.

[14]

X. FanB. Xu and J. Yang, Phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation, J. Differ. Equ., 266 (2019), 5821-5866.  doi: 10.1016/j.jde.2018.10.051.

[15]

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

[16]

C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation, Comm. Anal. Geom., 11 (2003), 223-233.  doi: 10.4310/CAG.2003.v11.n2.a3.

[17]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. R. Soc. Edinb. Sect. A Math., 111 (1989), 69-84.  doi: 10.1017/S0308210500025026.

[18]

M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions, Ann. Mat. Pura Appl., 184 (2005), 17-52.  doi: 10.1007/s10231-003-0088-y.

[19]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains, Discrete Contin. Dyn. Syst.-A, 32 (2012), 1391-1420.  doi: 10.3934/dcds.2012.32.1391.

[20]

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

[21]

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

[22]

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

[23]

F. Morgan, Manifolds with Density, Notices Amer. Math. Soc., 52 (2005), 853-858. 

[24]

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

[25]

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

[26]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differ. Geom., 64 (2003), 359-423.  doi: 10.4310/jdg/1090426999.

[27]

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

[28]

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

[29]

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

[30]

K. Sakamoto, Existence and stability of three-dimensional boundary-interior layers for the Allen-Cahn equation, Taiwan. J. Math., 9 (2005), 331-358.  doi: 10.11650/twjm/1500407844.

[31]

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

[32]

F. TangS. Wei and J. Yang, Phase transition layers for Fife-Greenlee problem on smooth bounded domain, Discrete Contin. Dyn. Syst.-A, 38 (2018), 1527-1552.  doi: 10.3934/dcds.2018063.

[33]

J. Wei and J. Yang, Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain, Discrete Contin. Dyn. Syst.-A, 22 (2008), 465-508. doi: 10.3934/dcds.2008.22.465.

[34]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model, Asymptotic Anal., 69 (2010), 175-218.  doi: 10.3233/ASY-2010-0999.

[35]

S. Wei, B. Xu and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains, Calc. Var. Partial Differ. Equ., 57 (2018), Article: 87. doi: 10.1007/s00526-018-1347-5.

[36]

S. Wei and J. Yang, Connectivity of boundaries by clustering phase transition layers of Fife-Greenlee problem on smooth bounded domain, J. Differ. Equ., to appear. doi: 10.1016/j.jde.2020.01.014.

[37]

J. Yang and X. Yang, Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains, Commun. Pure Appl. Anal., 12 (2013), 303-340.  doi: 10.3934/cpaa.2013.12.303.

Figure 1.  $\begin{array}{*{20}{l}} {{ Curves}{\rm{: }}{C_1} = \left( {{y_1},{\varphi _1}\left( {{y_1}} \right)} \right),\;\;\;{\kern 1pt} {C_2} = \left( {{y_1},{\varphi _2}\left( {{y_1}} \right)} \right), - {\delta _0} < {y_1} < {\delta _0},}\\ {{ Points}{\rm{: }}{P_1} = (0,\varphi (0)),{P_2} = \left( {0,{\varphi _2}(0)} \right).} \end{array}$
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