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 $Γ$.
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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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. del Pino
, Layers with nonsmooth interface in a semilinear elliptic problem, Comm. Partial Differential Equations, 17 (1992)
, 1695-1708.
doi: 10.1080/03605309208820900.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. P. do Carmo, Differential Geometry of Curves and Surfaces, Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1976.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
J. Hale
and K. Sakamoto
, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988)
, 367-405.
doi: 10.1007/BF03167908.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
L. Modica
, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987)
, 123-142.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
P. H. Rabinowitz
and E. Stredulinsky
, Mixed states for an Allen-Cahn type equation, Ⅱ, Calc. Var. Partial Differential Equations, 21 (2004)
, 157-207.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
S. Wei, B. Xu and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains, preprint, arXiv:1603.07175.
![]() |
|
J. Wei
and J. Yang
, Toda system and cluster phase transition layers in an inhomogeneous phase transition model, Asymptot. Anal., 69 (2010)
, 175-218.
![]() ![]() |
|
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.![]() ![]() ![]() |