Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$

Michał Kowalczyk; Yong Liu; Frank Pacard

doi:10.3934/nhm.2012.7.837

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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 $\mathbb{R}^2$

Michał Kowalczyk ^1, , Yong Liu ^2, and Frank Pacard ^3,

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

Abstract
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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.

Keywords: classification of solutions., moduli spaces, multiple-end solutions, Allen-Cahn equation.

Mathematics Subject Classification: Primary: 35B08, 35Q8.

Citation: Michał Kowalczyk, Yong Liu, Frank Pacard. Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$. Networks & 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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math., 62 (1988), 83-114. doi: 10.1007/BF01258269. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	M. Kowalczyk, Y. Liu and F. Pacard, The classification of four ended solutions to the Allen-Cahn equation on the plane, preprint, 2011. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math., 62 (1988), 83-114. doi: 10.1007/BF01258269. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	M. Kowalczyk, Y. Liu and F. Pacard, The classification of four ended solutions to the Allen-Cahn equation on the plane, preprint, 2011. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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