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

Feifei Tang; Suting Wei; Jun Yang

doi:10.3934/dcds.2018063

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

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

Feifei Tang ^1, , Suting Wei ^1, and Jun Yang ^2,,

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

$-1\,<\,\mathbf{a}(y)\,<1$

$Ω$

and

$\triangledown\mathbf{a}≠ 0$

$Γ$

, 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

$Γ$

Keywords: Fife-Greenlee problem, phase transition, resonance, fermi coordinates, boundary connectivity.

Mathematics Subject Classification: Primary: 35B34; Secondary: 35J25.

Citation: Feifei Tang, Suting Wei, Jun Yang. Phase transition layers for Fife-Greenlee problem on smooth bounded domain. Discrete and Continuous Dynamical Systems, 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	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.
[14]	M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal., 38 (2006/07), 1542-1564.
[15]	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.
[16]	M. P. do Carmo, Differential Geometry of Curves and Surfaces, Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1976.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	J. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405. doi: 10.1007/BF03167908.
[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.
[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.
[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.
[27]	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.
[28]	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.
[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.
[30]	L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[40]	S. Wei, B. Xu and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains, preprint, arXiv:1603.07175.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	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.
[14]	M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal., 38 (2006/07), 1542-1564.
[15]	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.
[16]	M. P. do Carmo, Differential Geometry of Curves and Surfaces, Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1976.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	J. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405. doi: 10.1007/BF03167908.
[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.
[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.
[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.
[27]	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.
[28]	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.
[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.
[30]	L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[40]	S. Wei, B. Xu and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains, preprint, arXiv:1603.07175.
[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.
[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.

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