Nondegeneracy of the saddle solution of the Allen-Cahn equation
HTML articles powered by AMS MathViewer
- by Michał Kowalczyk and Yong Liu
- Proc. Amer. Math. Soc. 139 (2011), 4319-4329
- DOI: https://doi.org/10.1090/S0002-9939-2011-11217-6
- Published electronically: July 26, 2011
- PDF | Request permission
Abstract:
A solution of the Allen-Cahn equation in the plane is called a saddle solution if its nodal set coincides with the coordinate axes. Such solutions are known to exist for a large class of nonlinearities. In this paper we consider the linear operator obtained by linearizing the Allen-Cahn equation around the saddle solution. Our result states that there are no nontrivial, decaying elements in the kernel of this operator. In other words, the saddle solution of the Allen-Cahn equation is nondegenerate.References
- Francesca Alessio, Alessandro Calamai, and Piero Montecchiari, Saddle-type solutions for a class of semilinear elliptic equations, Adv. Differential Equations 12 (2007), no. 4, 361–380. MR 2305872
- Henri Berestycki, Luis Caffarelli, and Louis Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69–94 (1998). Dedicated to Ennio De Giorgi. MR 1655510
- X. Cabré and J. Terra, Qualitative properties of saddle-shaped solutions to bistable diffusion equations, Comm. Partial Differential Equations 35 (2010), no. 11, 1923–1957. MR 2754074, DOI 10.1080/03605302.2010.484039
- Xavier Cabré and Joana Terra, Saddle-shaped solutions of bistable diffusion equations in all of $\Bbb R^{2m}$, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 4, 819–843. MR 2538506, DOI 10.4171/JEMS/168
- E. N. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann. 285 (1989), no. 4, 647–669. MR 1027764, DOI 10.1007/BF01452052
- Ha Dang, Paul C. Fife, and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys. 43 (1992), no. 6, 984–998. MR 1198672, DOI 10.1007/BF00916424
- M. del Pino, M. Kowalczyk, and F. Pacard. Moduli space theory for the Allen-Cahn equation in the plane. To appear in Trans. Amer. Math. Soc.
- Manuel del Pino, MichałKowalczyk, Frank Pacard, and Juncheng Wei, Multiple-end solutions to the Allen-Cahn equation in $\Bbb R^2$, J. Funct. Anal. 258 (2010), no. 2, 458–503. MR 2557944, DOI 10.1016/j.jfa.2009.04.020
- D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 (1985), no. 1, 121–132. MR 808112, DOI 10.1007/BF01394782
- N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481–491. MR 1637919, DOI 10.1007/s002080050196
- Changfeng Gui, Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal. 254 (2008), no. 4, 904–933. MR 2381198, DOI 10.1016/j.jfa.2007.10.015
- Philip Hartman and Aurel Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math. 75 (1953), 449–476. MR 58082, DOI 10.2307/2372496
- Joaquín Pérez and Martin Traizet, The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends, Trans. Amer. Math. Soc. 359 (2007), no. 3, 965–990. MR 2262839, DOI 10.1090/S0002-9947-06-04094-3
- M. Schatzman, On the stability of the saddle solution of Allen-Cahn’s equation, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 6, 1241–1275. MR 1363002, DOI 10.1017/S0308210500030493
Bibliographic Information
- Michał Kowalczyk
- Affiliation: Departamento de Ingeniería Matemática and CMM (UMI 2807, CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
- Email: kowalczy@dim.uchile.cl
- Yong Liu
- Affiliation: Departamento de Ingeniería Matemática and CMM (UMI 2807, CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
- Address at time of publication: School of Mathematics and Physics, North China Electric Power University, Beijing, People’s Republic of China 102206
- Email: yliu@dim.uchile.cl
- Received by editor(s): September 17, 2010
- Published electronically: July 26, 2011
- Additional Notes: The first author was partially supported by Chilean research grants Fondecyt 1090103, Fondo Basal CMM-Chile, and Project Añillo ACT-125 CAPDE
The second author was partially supported by Chilean research grants Fondecyt 3100011 and Fondo Basal CMM-Chile and doctoral grants of North China Electric Power University - Communicated by: Yingfei Yi
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 4319-4329
- MSC (2010): Primary 35B08, 35P99, 35Q80
- DOI: https://doi.org/10.1090/S0002-9939-2011-11217-6
- MathSciNet review: 2823077