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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. 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.  doi: 10.2307/2154670. [2] N. D. Alikakos, X. 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 Pino, M. 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 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. [9] M. del Pino, M. 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 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. [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. Fan, B. 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. Malchiodi, W. 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. Tang, S. 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. 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.  doi: 10.2307/2154670. [2] N. D. Alikakos, X. 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 Pino, M. 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 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. [9] M. del Pino, M. 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 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. [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. Fan, B. 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. Malchiodi, W. 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. Tang, S. 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.
$\begin{array}{*{20}{l}} {{ Curves}{\rm{: }}{C_1} = \left( {{y_1},{\varphi _1}\left( {{y_1}} \right)} \right),\;\;\;{\kern 1pt} {C_2} = \left( {{y_1},{\varphi _2}\left( {{y_1}} \right)} \right), - {\delta _0} < {y_1} < {\delta _0},}\\ {{ Points}{\rm{: }}{P_1} = (0,\varphi (0)),{P_2} = \left( {0,{\varphi _2}(0)} \right).} \end{array}$
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