March  2018, 38(3): 1527-1552. doi: 10.3934/dcds.2018063

Phase transition layers for Fife-Greenlee problem on smooth bounded domain

1. 

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

2. 

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

* Corresponding author: Jun Yang

Received  June 2017 Revised  September 2017 Published  December 2017

Fund Project: The third author is supported by NSFC(No. 11371254 and No. 11671144).

We consider the Fife-Greenlee problem
$ε^2\triangle u + \bigl(u-\mathbf{a}(y)\bigr)(1-u^2) =0 ~~~ \mbox{in}\ Ω,~~~~~~~\frac{\partial u}{\partialν} = 0 ~~~ \mbox{on}\ \partialΩ,$
where
$Ω$
is a bounded domain in
${\mathbb R}^2$
with smooth boundary,
$\epsilon>0$
is a small parameter,
$ν$
denotes the unit outward normal of
$\partialΩ$
. Let
$Γ = \{y∈ Ω: \mathbf{a}(y) = 0 \}$
be a simple smooth curve intersecting orthogonally with
$\partialΩ$
at exactly two points and dividing
$Ω$
into two disjoint nonempty components. We assume that
$-1\,<\,\mathbf{a}(y)\,<1$
on
$Ω$
and
$\triangledown\mathbf{a}≠ 0$
on
$Γ$
, and also some admissibility conditions between the curves
$Γ$
,
$\partialΩ$
and the inhomogeneity
${\mathbf a}$
hold at the connecting points. We can prove that there exists a solution
$u_{\epsilon}$
such that: as
$\epsilon → 0$
,
$u_{\epsilon}$
approaches
$+1$
in one part, while tends to
$-1$
in the other part, except a small neighborhood of
$Γ$
.
Citation: Feifei Tang, Suting Wei, Jun Yang. Phase transition layers for Fife-Greenlee problem on smooth bounded domain. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1527-1552. doi: 10.3934/dcds.2018063
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.  doi: 10.1016/S0294-1449(16)30349-3.  Google Scholar

[2]

N. D. AlikakosP. 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. 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.   Google Scholar

[4]

N. D. AlikakosX. 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. AngenentJ. 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.  doi: 10.4310/MRL.1996.v3.n1.a4.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

M. del PinoM. 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

[14]

M. del PinoM. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal., 38 (2006/07), 1542-1564.   Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

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

[18]

Z. Du and J. Wei, Clustering layers for the Fife-Greenlee problem in ${\mathbb R}^n$, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 107-139.  doi: 10.1017/S0308210515000360.  Google Scholar

[19]

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

[20]

P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter, Russian Math. Survey, 29 (1974), 103-130.   Google Scholar

[21]

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

[22]

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.  Google Scholar

[23]

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

[24]

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

[25]

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.  Google Scholar

[26]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operator, Mathematics and its application (Soviet Series), 59. Kluwer Acadamic Publishers Group, Dordrecht, 1991.  Google Scholar

[27]

F. MahmoudiA. 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

[28]

A. MalchiodiW.-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

[29]

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.  Google Scholar

[30]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.   Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, Ⅱ, Calc. Var. Partial Differential Equations, 21 (2004), 157-207.   Google Scholar

[37]

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

[38]

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

[39]

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

[40]

S. Wei, B. Xu and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains, preprint, arXiv:1603.07175. Google Scholar

[41]

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

[42]

J. Wei and J. Yang, Concentration on lines for a singularly perturbed Neumann problem in two-dimensional domains, Indiana Univ. Math. J., 56 (2007), 3025-3073.  doi: 10.1512/iumj.2007.56.3133.  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.  doi: 10.1016/S0294-1449(16)30349-3.  Google Scholar

[2]

N. D. AlikakosP. 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. 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.   Google Scholar

[4]

N. D. AlikakosX. 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. AngenentJ. 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.  doi: 10.4310/MRL.1996.v3.n1.a4.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

M. del PinoM. 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

[14]

M. del PinoM. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal., 38 (2006/07), 1542-1564.   Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

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

[18]

Z. Du and J. Wei, Clustering layers for the Fife-Greenlee problem in ${\mathbb R}^n$, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 107-139.  doi: 10.1017/S0308210515000360.  Google Scholar

[19]

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

[20]

P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter, Russian Math. Survey, 29 (1974), 103-130.   Google Scholar

[21]

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

[22]

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.  Google Scholar

[23]

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

[24]

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

[25]

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.  Google Scholar

[26]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operator, Mathematics and its application (Soviet Series), 59. Kluwer Acadamic Publishers Group, Dordrecht, 1991.  Google Scholar

[27]

F. MahmoudiA. 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

[28]

A. MalchiodiW.-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

[29]

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.  Google Scholar

[30]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.   Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, Ⅱ, Calc. Var. Partial Differential Equations, 21 (2004), 157-207.   Google Scholar

[37]

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

[38]

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

[39]

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

[40]

S. Wei, B. Xu and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains, preprint, arXiv:1603.07175. Google Scholar

[41]

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

[42]

J. Wei and J. Yang, Concentration on lines for a singularly perturbed Neumann problem in two-dimensional domains, Indiana Univ. Math. J., 56 (2007), 3025-3073.  doi: 10.1512/iumj.2007.56.3133.  Google Scholar

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