Abstract
For a solution of the Allen–Cahn equation in \({\mathbb R}^2\), under the natural linear growth energy bound, we show that the blowing down limit is unique. Furthermore, if the solution has finite Morse index, the blowing down limit satisfies the multiplicity one property.
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Devyver, B.: On the finiteness of the Morse index for Schrödinger operators. Manuscripta Math. 139(1-2), 249–271 (2012)
del Pino, M., Kowalczyk, M., Pacard, F.: Moduli space theory for the Allen–Cahn equation in the plane. Trans. Am. Math. Soc. 365(2), 721–766 (2013)
del Pino, M., Kowalczyk, M., Wei, J., Yang, J.: Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature. Geom. Funct. Anal. 20, 918–957 (2010)
Gui, C.: Hamiltonian identities for elliptic partial differential equations. J. Funct. Anal. 254(4), 904–933 (2008)
Gui, C.: Symmetry of some entire solutions to the Allen–Cahn equation in two dimensions. J. Differ. Equ. 252(11), 5853–5874 (2012)
Hutchinson, J., Tonegawa, Y.: Convergence of phase interfaces in the van der Waals–Cahn–Hilliard theory. Calc. Var. PDEs 10(1), 49–84 (2000)
Kowalczyk, M., Liu, Y., Pacard, F.: The space of 4-ended solutions to the Allen–Cahn equation in the plane. Ann. Inst. H. Poincaré Anal. Non Linéaire 29(5), 761–781 (2012)
Kowalczyk, M., Liu, Y., Pacard, F.: The classification of four-end solutions to the Allen–Cahn equation on the plane. Anal. PDE 6(7), 1675–1718 (2013)
Kowalczyk, M., Liu, Y., Pacard, F.: Towards classification of multiple-end solutions to the Allen–Cahn equation in \({\mathbb{R}}^2\). Netw. Heterog. Media 7(4), 837–855 (2013)
Modica, L.: A gradient bound and a Liouville theorem for nonlinear Poisson equations. Comm. Pure Appl. Math. 38(5), 679–684 (1985)
Smyrnelis, P.: Gradient estimates for semilinear elliptic systems and other related results. In: Proceedings of the Royal Society of Edinburgh: Section A Mathematics, available on CJO2015 (Accepted in Journal of the European Mathematical Society)
Tonegawa, Y.: On stable critical points for a singular perturbation problem. Commun. Anal. Geom. 13(2), 439–459 (2005)
Wang, K., Wei, J.: Finite Morse index implies finite ends. arXiv:1705.06831
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Wang, K. Some remarks on the structure of finite Morse index solutions to the Allen–Cahn equation in \({\mathbb R}^2\) . Nonlinear Differ. Equ. Appl. 24, 58 (2017). https://doi.org/10.1007/s00030-017-0481-7
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DOI: https://doi.org/10.1007/s00030-017-0481-7