Multiple end solutions to the Allen-Cahn equation in 2
Séminaire Laurent Schwartz — EDP et applications (2013-2014), Exposé no. 10, 19 p.

An entire solution of the Allen-Cahn equation Δu=f(u), where f is an odd function and has exactly three zeros at ±1 and 0, e.g. f(u)=u(u 2 -1), is called a 2k end 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 this solutions with just four ends. A special example is the saddle solutions U whose nodal lines are precisely the straight lines y=±x. We describe completely connected components of the moduli space of four end solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all four end solutions are continuous deformations of the saddle solution.

DOI : 10.5802/slsedp.55
Kowalczyk, Michał 1 ; Liu, Yong 1 ; Pacard, Frank 2

1 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS) Universidad de Chile Casilla 170 Correo 3 Santiago Chile
2 Centre de Mathématiques Laurent Schwartz and Institut Universitaire de France École Polytechnique 91128 Palaiseau France
@article{SLSEDP_2013-2014____A10_0,
     author = {Kowalczyk, Micha{\l} and Liu, Yong and Pacard, Frank},
     title = {Multiple end solutions to the {Allen-Cahn} equation in $\mathbb{R}^2$},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:10},
     pages = {1--19},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2013-2014},
     doi = {10.5802/slsedp.55},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/slsedp.55/}
}
TY  - JOUR
AU  - Kowalczyk, Michał
AU  - Liu, Yong
AU  - Pacard, Frank
TI  - Multiple end solutions to the Allen-Cahn equation in $\mathbb{R}^2$
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:10
PY  - 2013-2014
SP  - 1
EP  - 19
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://archive.numdam.org/articles/10.5802/slsedp.55/
DO  - 10.5802/slsedp.55
LA  - en
ID  - SLSEDP_2013-2014____A10_0
ER  - 
%0 Journal Article
%A Kowalczyk, Michał
%A Liu, Yong
%A Pacard, Frank
%T Multiple end solutions to the Allen-Cahn equation in $\mathbb{R}^2$
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:10
%D 2013-2014
%P 1-19
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://archive.numdam.org/articles/10.5802/slsedp.55/
%R 10.5802/slsedp.55
%G en
%F SLSEDP_2013-2014____A10_0
Kowalczyk, Michał; Liu, Yong; Pacard, Frank. Multiple end solutions to the Allen-Cahn equation in $\mathbb{R}^2$. Séminaire Laurent Schwartz — EDP et applications (2013-2014), Exposé no. 10, 19 p. doi : 10.5802/slsedp.55. http://archive.numdam.org/articles/10.5802/slsedp.55/

[1] F. Alessio, A. Calamai, and P. Montecchiari. Saddle-type solutions for a class of semilinear elliptic equations. Adv. Differential Equations, 12(4):361–380, 2007. | MR | Zbl

[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(4):725–739 (electronic), 2000. | MR | Zbl

[3] M. T. Barlow, R. F. Bass, and C. Gui. The Liouville property and a conjecture of De Giorgi. Comm. Pure Appl. Math., 53(8):1007–1038, 2000. | MR | Zbl

[4] H. Berestycki, F. Hamel, and R. Monneau. One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J., 103(3):375–396, 2000. | MR | Zbl

[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(3):1225–1243 (electronic), 2005. | MR | Zbl

[6] H. Dang, P. C. Fife, and L. A. Peletier. Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys., 43(6):984–998, 1992. | MR | Zbl

[7] M. del Pino, M. Kowalczyk, and F. Pacard. Moduli space theory for the Allen-Cahn equation in the plane. to appear Transactions AMS, 2010. | MR | Zbl

[8] M. del Pino, M. Kowalczyk, F. Pacard, and J. Wei. Multiple-end solutions to the Allen-Cahn equation in 2 . J. Funct. Anal., 258(2):458–503, 2010. | MR | Zbl

[9] M. del Pino, M. Kowalczyk, and J. Wei. On De Giorgi’s in dimension N9. Ann. of Math. (2), 174(3):1485–1569, 2011. | MR | Zbl

[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(4):255–265, 1999. | MR | Zbl

[11] D. Fischer-Colbrie. On complete minimal surfaces with finite Morse index in three-manifolds. Invent. Math., 82(1):121–132, 1985. | MR | Zbl

[12] N. Ghoussoub and C. Gui. On a conjecture of De Giorgi and some related problems. Math. Ann., 311(3):481–491, 1998. | MR | Zbl

[13] C. Gui. Hamiltonian identities for elliptic partial differential equations. J. Funct. Anal., 254(4):904–933, 2008. | MR | Zbl

[14] C. Gui. Even Symmetry of Some Entire Solutions to the Allen-Cahn Equation in Two Dimensions. J. Differential Equations, 252(11):5853–5874, 2012. | MR | Zbl

[15] H. Karcher. Embedded minimal surfaces derived from Scherk’s examples. Manuscripta Math., 62(1):83–114, 1988. | MR | Zbl

[16] B. Kostant. The solution to a generalized Toda lattice and representation theory. Adv. in Math., 34(3):195–338, 1979. | MR | Zbl

[17] M. Kowalczyk and Y. Liu. Nondegeneracy of the saddle solution of the Allen-Cahn equation. Proc. Amer. Math. Soc., 139(12):43–4329, 2011. | MR | Zbl

[18] M. Kowalczyk, Y. Liu, and F. Pacard. The classification of four ended solutions to the Allen-Cahn equation on the plane. preprint, 2011. | MR | Zbl

[19] M. Kowalczyk, Y. Liu, and F. Pacard. The space of four ended solutions to the Allen-Cahn equation on the plane. Ann. Inst. H. Poincaré Anal. Non Linéaire, 29(5):761–781, 2012. | Numdam | MR | Zbl

[20] R. Kusner, R. Mazzeo, and D. Pollack. The moduli space of complete embedded constant mean curvature surfaces. Geom. Funct. Anal., 6(1):120–137, 1996. | MR | Zbl

[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, pages 17–51. Cambridge Univ. Press, Cambridge, 1998. | MR | Zbl

[22] R. Mazzeo, D. Pollack, and K. Uhlenbeck. Moduli spaces of singular Yamabe metrics. J. Amer. Math. Soc., 9(2):303–344, 1996. | MR | Zbl

[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(2):441–465, 2007. | MR | Zbl

[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), pages 467–497. Lecture Notes in Phys., Vol. 38. Springer, Berlin, 1975. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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(3):965–990 (electronic), 2007. | MR | Zbl

[28] O. Savin. Regularity of flat level sets in phase transitions. Ann. of Math. (2), 169(1):41–78, 2009. | MR | Zbl

[29] M. Schatzman. On the stability of the saddle solution of Allen-Cahn’s equation. Proc. Roy. Soc. Edinburgh Sect. A, 125(6):1241–1275, 1995. | MR | Zbl

Cité par Sources :