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Existence of nonnegative solutions to singular elliptic problems, a variational approach
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 |
$ε^2\triangle u + \bigl(u-\mathbf{a}(y)\bigr)(1-u^2) =0 ~~~ \mbox{in}\ Ω,~~~~~~~\frac{\partial u}{\partialν} = 0 ~~~ \mbox{on}\ \partialΩ,$ |
$Ω$ |
${\mathbb R}^2$ |
$\epsilon>0$ |
$ν$ |
$\partialΩ$ |
$Γ = \{y∈ Ω: \mathbf{a}(y) = 0 \}$ |
$\partialΩ$ |
$Ω$ |
$-1\,<\,\mathbf{a}(y)\,<1$ |
$Ω$ |
$\triangledown\mathbf{a}≠ 0$ |
$Γ$ |
$Γ$ |
$\partialΩ$ |
${\mathbf a}$ |
$u_{\epsilon}$ |
$\epsilon → 0$ |
$u_{\epsilon}$ |
$+1$ |
$-1$ |
$Γ$ |
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. 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.
|
[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. 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.
|
[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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