# American Institute of Mathematical Sciences

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.

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##### 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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