December  2012, 7(4): 837-855. doi: 10.3934/nhm.2012.7.837

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

Received  May 2012 Revised  October 2012 Published  December 2012

An entire solution of the Allen-Cahn equation $\Delta u=f(u)$, where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$, e.g. $f(u)=u(u^2-1)$, is called a $2k$-ended solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $-1$) like the one dimensional, odd, heteroclinic solution $H$, of $H''=f(H)$. In this paper we present some recent advances in the theory of the multiple-end solutions. We begin with the description of the moduli space of such solutions. Next we move on to study a special class of these solutions with just four ends. A special example is the saddle solutions $U$ whose nodal lines are precisely the straight lines $y=\pm x$. We describe the connected components of the moduli space of $4$-ended solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all $4$-ended solutions are continuous deformations of the saddle solution.
Citation: Michał Kowalczyk, Yong Liu, Frank Pacard. Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$. Networks and Heterogeneous Media, 2012, 7 (4) : 837-855. doi: 10.3934/nhm.2012.7.837
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 $\mathbb{R}^2$, 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 $\mathbb{R}^2$, 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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