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Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$
1. | Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile |
2. | Departamento de Ingeniería Matemática and CMM, (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago |
3. | Centre de Mathématiques Laurent Schwartz, École Polytechnique, UMR-CNRS 7640, 91128 Palaiseau, France |
References:
[1] |
F. Alessio, A. Calamai and P. Montecchiari, Saddle-type solutions for a class of semilinear elliptic equations, Adv. Differential Equations, 12 (2007), 361-380. |
[2] |
L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $R^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739. (electronic)
doi: 10.1090/S0894-0347-00-00345-3. |
[3] |
M. T. Barlow, R. F. Bass and C. Gui, The Liouville property and a conjecture of De Giorgi, Comm. Pure Appl. Math., 53 (2000), 1007-1038.
doi: 10.1002/1097-0312(200008)53:8<1007::AID-CPA3>3.3.CO;2-L. |
[4] |
H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.
doi: 10.1215/S0012-7094-00-10331-6. |
[5] |
E. N. Dancer, Stable and finite Morse index solutions on $ R^n$or on bounded domains with small diffusion , Trans. Amer. Math. Soc., 357 (2005), 1225-1243. (electronic).
doi: 10.1090/S0002-9947-04-03543-3. |
[6] |
H. Dang, P. C. Fife and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys., 43 (1992), 984-998.
doi: 10.1007/BF00916424. |
[7] |
M. del Pino, M. Kowalczyk and F. Pacard, Moduli space theory for the Allen-Cahn equation in the plane, Trans. Amer. Math. Soc., 365 (2013), 721-766.
doi: 10.1090/S0002-9947-2012-05594-2. |
[8] |
M. del Pino, M. Kowalczyk, F. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $\mathbbR^2$, J. Funct. Anal., 258 (2010), 458-503.
doi: 10.1016/j.jfa.2009.04.020. |
[9] |
M. del Pino, M. Kowalczyk and J. Wei, On De Giorgi's in dimension $N\geq 9$, Ann. of Math. (2), 174 (2011), 1485-1569.
doi: 10.4007/annals.2011.174.3.3. |
[10] |
A. Farina, Symmetry for solutions of semilinear elliptic equations in $R^N$ and related conjectures, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 10 (1999), 255-265. |
[11] |
D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math., 82 (1985), 121-132.
doi: 10.1007/BF01394782. |
[12] |
N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491.
doi: 10.1007/s002080050196. |
[13] |
C. Gui, Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal., 254 (2008), 904-933.
doi: 10.1016/j.jfa.2007.10.015. |
[14] |
C. Gui, Even symmetry of some entire solutions to the Allen-Cahn equation in two dimensions, J. Differential Equations, 252 (2012), 5853-5874.
doi: 10.1016/j.jde.2012.03.004. |
[15] |
H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math., 62 (1988), 83-114.
doi: 10.1007/BF01258269. |
[16] |
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math., 34 (1979), 195-338.
doi: 10.1016/0001-8708(79)90057-4. |
[17] |
M. Kowalczyk and Y. Liu, Nondegeneracy of the saddle solution of the Allen-Cahn equation, Proc. Amer. Math. Soc., 139 (2011), 43-4329.
doi: 10.1090/S0002-9939-2011-11217-6. |
[18] |
M. Kowalczyk, Y. Liu and F. Pacard, The classification of four ended solutions to the Allen-Cahn equation on the plane, preprint, 2011. |
[19] |
M. Kowalczyk, Y. Liu and F. Pacard, The space of 4-ended solutions to the Allen-Cahn equation in the plane, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 761-781.
doi: 10.1016/j.anihpc.2012.04.003. |
[20] |
R. Kusner, R. Mazzeo and D. Pollack, The moduli space of complete embedded constant mean curvature surfaces, Geom. Funct. Anal., 6 (1996), 120-137.
doi: 10.1007/BF02246769. |
[21] |
R. Mazzeo and D. Pollack, Gluing and moduli for noncompact geometric problems, in "Geometric Theory of Singular Phenomena in Partial Differential Equations (Cortona, 1995)", Sympos. Math., XXXVIII, 17-51. Cambridge Univ. Press, Cambridge, (1998). |
[22] |
R. Mazzeo, D. Pollack and K. Uhlenbeck, Moduli spaces of singular Yamabe metrics, J. Amer. Math. Soc., 9 (1996), 303-344.
doi: 10.1090/S0894-0347-96-00208-1. |
[23] |
W. H. Meeks, III and M. Wolf, Minimal surfaces with the area growth of two planes: The case of infinite symmetry, J. Amer. Math. Soc., 20 (2007), 441-465.
doi: 10.1090/S0894-0347-06-00537-6. |
[24] |
J. Moser, Finitely many mass points on the line under the influence of an exponential potential-an integrable system, in "Dynamical Systems, Theory and Applications (Rencontres, BattelleRes. Inst., Seattle, Wash., 1974)", 467-497. Lecture Notes in Phys., 38. Springer, Berlin, (1975). |
[25] |
A. F. Nikiforov and V. B. Uvarov, "Special Functions of Mathematical Physics," Birkhäuser Verlag, Basel, 1988. A unified introduction with applications, Translated from the Russian and with a preface by Ralph P. Boas, With a foreword by A. A. Samarskiĭ. |
[26] |
F. Pacard and J. Wei, Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, J. Funct. Anal. to appear, (2011). |
[27] |
J. Pérez and M. Traizet, The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends, Trans. Amer. Math. Soc., 359 (2007), 965-990. (electronic).
doi: 10.1090/S0002-9947-06-04094-3. |
[28] |
O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2), 169 (2009), 41-78.
doi: 10.4007/annals.2009.169.41. |
[29] |
M. Schatzman, On the stability of the saddle solution of Allen-Cahn's equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1241-1275.
doi: 10.1017/S0308210500030493. |
show all references
References:
[1] |
F. Alessio, A. Calamai and P. Montecchiari, Saddle-type solutions for a class of semilinear elliptic equations, Adv. Differential Equations, 12 (2007), 361-380. |
[2] |
L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $R^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739. (electronic)
doi: 10.1090/S0894-0347-00-00345-3. |
[3] |
M. T. Barlow, R. F. Bass and C. Gui, The Liouville property and a conjecture of De Giorgi, Comm. Pure Appl. Math., 53 (2000), 1007-1038.
doi: 10.1002/1097-0312(200008)53:8<1007::AID-CPA3>3.3.CO;2-L. |
[4] |
H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.
doi: 10.1215/S0012-7094-00-10331-6. |
[5] |
E. N. Dancer, Stable and finite Morse index solutions on $ R^n$or on bounded domains with small diffusion , Trans. Amer. Math. Soc., 357 (2005), 1225-1243. (electronic).
doi: 10.1090/S0002-9947-04-03543-3. |
[6] |
H. Dang, P. C. Fife and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys., 43 (1992), 984-998.
doi: 10.1007/BF00916424. |
[7] |
M. del Pino, M. Kowalczyk and F. Pacard, Moduli space theory for the Allen-Cahn equation in the plane, Trans. Amer. Math. Soc., 365 (2013), 721-766.
doi: 10.1090/S0002-9947-2012-05594-2. |
[8] |
M. del Pino, M. Kowalczyk, F. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $\mathbbR^2$, J. Funct. Anal., 258 (2010), 458-503.
doi: 10.1016/j.jfa.2009.04.020. |
[9] |
M. del Pino, M. Kowalczyk and J. Wei, On De Giorgi's in dimension $N\geq 9$, Ann. of Math. (2), 174 (2011), 1485-1569.
doi: 10.4007/annals.2011.174.3.3. |
[10] |
A. Farina, Symmetry for solutions of semilinear elliptic equations in $R^N$ and related conjectures, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 10 (1999), 255-265. |
[11] |
D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math., 82 (1985), 121-132.
doi: 10.1007/BF01394782. |
[12] |
N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491.
doi: 10.1007/s002080050196. |
[13] |
C. Gui, Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal., 254 (2008), 904-933.
doi: 10.1016/j.jfa.2007.10.015. |
[14] |
C. Gui, Even symmetry of some entire solutions to the Allen-Cahn equation in two dimensions, J. Differential Equations, 252 (2012), 5853-5874.
doi: 10.1016/j.jde.2012.03.004. |
[15] |
H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math., 62 (1988), 83-114.
doi: 10.1007/BF01258269. |
[16] |
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math., 34 (1979), 195-338.
doi: 10.1016/0001-8708(79)90057-4. |
[17] |
M. Kowalczyk and Y. Liu, Nondegeneracy of the saddle solution of the Allen-Cahn equation, Proc. Amer. Math. Soc., 139 (2011), 43-4329.
doi: 10.1090/S0002-9939-2011-11217-6. |
[18] |
M. Kowalczyk, Y. Liu and F. Pacard, The classification of four ended solutions to the Allen-Cahn equation on the plane, preprint, 2011. |
[19] |
M. Kowalczyk, Y. Liu and F. Pacard, The space of 4-ended solutions to the Allen-Cahn equation in the plane, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 761-781.
doi: 10.1016/j.anihpc.2012.04.003. |
[20] |
R. Kusner, R. Mazzeo and D. Pollack, The moduli space of complete embedded constant mean curvature surfaces, Geom. Funct. Anal., 6 (1996), 120-137.
doi: 10.1007/BF02246769. |
[21] |
R. Mazzeo and D. Pollack, Gluing and moduli for noncompact geometric problems, in "Geometric Theory of Singular Phenomena in Partial Differential Equations (Cortona, 1995)", Sympos. Math., XXXVIII, 17-51. Cambridge Univ. Press, Cambridge, (1998). |
[22] |
R. Mazzeo, D. Pollack and K. Uhlenbeck, Moduli spaces of singular Yamabe metrics, J. Amer. Math. Soc., 9 (1996), 303-344.
doi: 10.1090/S0894-0347-96-00208-1. |
[23] |
W. H. Meeks, III and M. Wolf, Minimal surfaces with the area growth of two planes: The case of infinite symmetry, J. Amer. Math. Soc., 20 (2007), 441-465.
doi: 10.1090/S0894-0347-06-00537-6. |
[24] |
J. Moser, Finitely many mass points on the line under the influence of an exponential potential-an integrable system, in "Dynamical Systems, Theory and Applications (Rencontres, BattelleRes. Inst., Seattle, Wash., 1974)", 467-497. Lecture Notes in Phys., 38. Springer, Berlin, (1975). |
[25] |
A. F. Nikiforov and V. B. Uvarov, "Special Functions of Mathematical Physics," Birkhäuser Verlag, Basel, 1988. A unified introduction with applications, Translated from the Russian and with a preface by Ralph P. Boas, With a foreword by A. A. Samarskiĭ. |
[26] |
F. Pacard and J. Wei, Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, J. Funct. Anal. to appear, (2011). |
[27] |
J. Pérez and M. Traizet, The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends, Trans. Amer. Math. Soc., 359 (2007), 965-990. (electronic).
doi: 10.1090/S0002-9947-06-04094-3. |
[28] |
O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2), 169 (2009), 41-78.
doi: 10.4007/annals.2009.169.41. |
[29] |
M. Schatzman, On the stability of the saddle solution of Allen-Cahn's equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1241-1275.
doi: 10.1017/S0308210500030493. |
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