# zbMATH — the first resource for mathematics

Symmetry of some entire solutions to the Allen-Cahn equation in two dimensions. (English) Zbl 1250.35078
In this paper, the author proves even symmetry and monotonicity of certain solutions of the Allen-Cahn equation $$u_{xx}+u_{yy}-F'(u)=0$$ in a half plane. The author also shows that entire solutions with finite Morse index and four ends must be evenly symmetric with respect to two orthogonal axes. A classification scheme of general entire solutions with finite Morse index is also presented using energy quantization.

##### MSC:
 35J20 Variational methods for second-order elliptic equations 35J60 Nonlinear elliptic equations 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 49Q05 Minimal surfaces and optimization 53A04 Curves in Euclidean and related spaces
Full Text:
##### References:
 [1] Alama, Stanley; Bronsard, Lia; Gui, Changfeng, Stationary layered solutions in $$R^2$$ for an Allen-Cahn systems with multiple-well potentials, Calc. var. partial differential equations, 5, 359-390, (1997) · Zbl 0883.35036 [2] Alessio, F.; Calamai, A.; Montecchiari, P., Saddle-type solutions for a class of semilinear elliptic equations, Adv. differential equations, 12, 4, 361-380, (2007) · Zbl 1193.35058 [3] Ambrosio, L.; Cabre, X., Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi, J. amer. math. soc., 13, 4, 725-739, (2000) · Zbl 0968.35041 [4] de Giorgi, E., Convergence problems for functionals and operators, () · Zbl 0405.49001 [5] Bethuel, F.; Brezis, H.; Helein, F., Ginzburg-Landau vortices, (1994), Birkhäuser Boston · Zbl 0802.35142 [6] Busca, J.; Felmer, P., Qualitative properties of some bounded positive solutions to scalar field equations, Calc. var., 13, 181-211, (2001) · Zbl 1151.35344 [7] Busca, J.; Jendoubi, M.-A.; Polacik, P., Convergence to equilibrium for semilinear parabolic problems in $$\mathbb{R}^n$$, Comm. partial differential equations, 27, 1793-1814, (2002) · Zbl 1021.35013 [8] Cabré, X.; Terra, J., Saddle-shaped solutions of bistable diffusion equations in all of $$\mathbb{R}^{2 m}$$, J. eur. math. soc. (JEMS), 11, 4, 819-843, (2009) · Zbl 1182.35110 [9] Cabré, X.; Terra, J., Qualitative properties of saddle-shaped solutions to bistable diffusion equations, Comm. partial differential equations, 35, 11, 1923-1957, (2010) · Zbl 1209.35042 [10] Dang, H.; Fife, P.C.; Peletier, L.A., Saddle solutions of the bistable diffusion equation, Z. angew. math. phys., 43, 6, 984-998, (1992) · Zbl 0764.35048 [11] M. del Pino, M. Kowalczyk, F. Pacard, Moduli space theory for the Allen-Cahn equation in the plane, Trans. Amer. Math. Soc. (2010), in press. · Zbl 1286.35018 [12] del Pino, M.; Kowalczyk, M.; Wei, J., On de giorgiʼs in dimension $$N \geqslant 9$$, Ann. of math. (2), 174, 3, 1485-1569, (2011) · Zbl 1238.35019 [13] del Pino, M.; Kowalczyk, M.; Pacard, F.; Wei, J., Multiple end solutions to the Allen-Cahn equation in $$\mathbb{R}^2$$, J. funct. anal., 258, 2, 458-503, (2010) · Zbl 1203.35108 [14] De Silva, D.; Savin, O., Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2D, Indiana univ. math. J., 58, 1, 301-315, (2009) · Zbl 1165.35021 [15] Fischer-Colbrie, D., On complete minimal surfaces with finite Morse index in three-manifolds, Invent. math., 82, 1, 121-132, (1985) · Zbl 0573.53038 [16] Ghoussoub, N.; Gui, C., On a conjecture of De Giorgi and some related problems, Math. ann., 311, 481-491, (1998) · Zbl 0918.35046 [17] Ghoussoub, N.; Gui, C., On de giorgiʼs conjecture in dimensions 4 and 5, Ann. of math., 157, 313-334, (2003) · Zbl 1165.35367 [18] C. Gui, Lecture Notes on Allen-Cahn type equations, in preparation, draft available at: www.math.uconn.edu/ gui. [19] Gui, C., Hamiltonian identity for elliptic partial differential equations, J. funct. anal., 254, 4, 904-933, (2008) · Zbl 1148.35023 [20] Gui, C., Symmetry of traveling wave solutions to the Allen-Cahn equation in $$\mathbb{R}^2$$, Arch. ration. mech. anal., 203, 3, 1037-1065, (2012) · Zbl 1256.35008 [21] Gui, C.; Malchiodi, A.; Xu, H., Axial of symmetry of stationary solutions to super critical nonlinear schrodinger equation, Proc. amer. math. soc., 139, 1023-1032, (2011) · Zbl 1211.35115 [22] Kowalczyk, M.; Liu, Y., Nondegeneracy of the saddle solution of the Allen-Cahn equation, Proc. amer. math. soc., 139, 12, 4319-4329, (2011) · Zbl 1241.35079 [23] M. Kowalczyk, Y. Liu, F. Pacard, The classification of four ended solutions to the Allen-Cahn equation on the plane, preprint, 2011. · Zbl 1287.35031 [24] M. Kowalczyk, Y. Liu, F. Pacard, The space of four ended solutions to the Allen-Cahn equation on the plane, preprint, 2011. · Zbl 1254.35219 [25] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, vol. II, 1951, translated from Russian. · Zbl 0043.19803 [26] Modica, L., A gradient bound and a Liouville theorem for non linear Poisson equations, Comm. pure appl. math., 38, 679-684, (1985) · Zbl 0612.35051 [27] Modica, L., Monotonicity of the energy for entire solutions of semilinear elliptic equations, () · Zbl 0699.35082 [28] Pérez, J.; Traizet, M., The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends, Trans. amer. math. soc., 359, 3, 965-990, (2007) · Zbl 1110.53008 [29] Savin, O., Regularity of at level sets in phase transitions, Ann. of math. (2), 169, 1, 41-78, (2009) · Zbl 1180.35499 [30] Schoen, R.M., The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. pure appl. math., 41, 3, 317-392, (1988) · Zbl 0674.35027 [31] Schatzman, M., On the stability of the saddle solution of Allen-cahnʼs equation, Proc. roy. soc. Edinburgh sect. A, 125, 6, 1241-1275, (1995) · Zbl 0852.35020 [32] Tonegawa, Y., On stable critical points for a singular perturbation problem, Comm. anal. geom., 13, 2, 439-459, (2005) · Zbl 1105.35008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.