February  2017, 37(2): 801-827. doi: 10.3934/dcds.2017033

Rotationally symmetric solutions to the Cahn-Hilliard equation

1. 

Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

2. 

Departamento de Ingeniería Matemática and Centro, de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Á. Hérnandez was partially supported by Chilean research grants Fondecyt 109103 and Fondo Basal CMM-Chile, Project Anillo ACT-125.M. Kowalczyk was partially supported by Chilean research grants Fondecyt 1090103,1130126, Fondo Basal CMM-Chile, Project Añillo ACT-125 CAPDE..The Cahn-Hilliard equation and related to it the Allen-Cahn equation and the phase field model have been a subject of extensive research of many mathematicians for more than 30 years. We have been a part of this group and we owe it to Pauf Fife whose papers in the early 90ties were for us an introduction to the area and an inspiration for the present work. For this reason we think it is appropriate to dedicate it to his memory.

Received  April 2015 Revised  November 2015 Published  November 2016

This paper is devoted to construction of new solutions to the Cahn-Hilliard equation in $\mathbb R^d$. Staring from the Delaunay unduloid $D_τ$ with parameter $τ∈ (0,τ^*)$ we find for each sufficiently small $\varepsilon $ a solution $u$ of this equation which is periodic in the direction of the $x_d$ axis and rotationally symmetric with respect to rotations about this axis. The zero level set of $u$ approaches as $\varepsilon \to 0$ the surface $D_τ$. We use a refined version of the Lyapunov-Schmidt reduction method which simplifies very technical aspects of previous constructions for similar problems.

Citation: Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
References:
[1]

N. D. AlikakosP. W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205.  doi: 10.1007/BF00375025.  Google Scholar

[2]

N. D. AlikakosX. Chen and G. Fusco, Motion of a droplet by surface tension along the boundary, Calc. Var. Partial Differential Equations, 11 (2000), 233-305.  doi: 10.1007/s005260000052.  Google Scholar

[3]

N.D. Alikakos and G. Fusco, The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions, Indiana Univ. Math. J., 42 (1993), 637-674.  doi: 10.1512/iumj.1993.42.42028.  Google Scholar

[4]

N.D. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: The motion of bubbles, Arch. Rational Mech. Anal., 141 (1998), 1-61.  doi: 10.1007/s002050050072.  Google Scholar

[5]

N.D. AlikakosG. Fusco and V. Stefanopoulos, Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, J. Differential Equations, 126 (1996), 106-167.  doi: 10.1006/jdeq.1996.0046.  Google Scholar

[6]

P.W. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations, 160 (2000), 283-356.  doi: 10.1006/jdeq.1999.3660.  Google Scholar

[7]

X. Chen, Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces, Comm. Partial Differential Equations, 19 (1994), 1371-1395.  doi: 10.1080/03605309408821057.  Google Scholar

[8]

X. Chen and M. Kowalczyk, Existence of equilibria for the Cahn-Hilliard equation via local minimizers of the perimeter, Comm. Partial Differential Equations, 21 (1996), 1207-1233.  doi: 10.1080/03605309608821223.  Google Scholar

[9]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, volume 251 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], SpringerVerlag, New York-Berlin, 1982. Google Scholar

[10]

H. DangP.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

[11]

M. del PinoM. KowalczykF. Pacard and J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516.  doi: 10.1016/j.aim.2010.01.003.  Google Scholar

[12]

M. del PinoM. Kowalczyk and J. Wei, On De Giorgi's conjecture in dimension N ≥ 9, Ann. of Math. (2)(7), 174 (2011), 1485-1569.  doi: 10.4007/annals.2011.174.3.3.  Google Scholar

[13]

M. del PinoM. Kowalczyk and J. Wei, Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in ${{\mathbb{R}}^{3}}$, Journ. Diff. Geometry, 93 (2013), 67-131.   Google Scholar

[14]

M. del PinoM. Kowalczyk and J. Wei, Traveling waves with multiple and non-convex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.  doi: 10.1002/cpa.21438.  Google Scholar

[15]

M. del PinoF. Pacard and M. Musso, Solutions of the Allen-Cahn equation which are invariant under screw-motion, Manuscripta Math., 138 (2012), 273-286.  doi: 10.1007/s00229-011-0492-3.  Google Scholar

[16]

C. Delaunay, Sur la surface de revolution dont la courbure moyenna est constante, J. Math. Pures Appl., 6 (1841), 309-320.   Google Scholar

[17]

J. Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57.  doi: 10.1007/BF03023575.  Google Scholar

[18]

P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations, 2000, pages No. 48, 26 pp. (electronic). Google Scholar

[19]

P.C. Fife, Pattern formation in gradient systems, In Handbook of dynamical systems, NorthHolland, Amsterdam, 2 (2002), 677-722.  doi: 10.1016/S1874-575X(02)80034-0.  Google Scholar

[20]

Á. Hernández and M. Kowalczyk, Delaunay end solutions of the cahn-hilliard equation in, in ´ preparation. Google Scholar

[21]

W.-y. Hsiang and W.C. Yu, A generalization of a theorem of Delaunay, J. Differential Geom., 16 (1981), 161-177.   Google Scholar

[22]

J.E. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der {W}aals-Cahn-Hilliard theory, Calc. Var. Partial Differential Equations, 10 (2000), 49-84.  doi: 10.1007/PL00013453.  Google Scholar

[23]

M. Jleli, End-to-end gluing of constant mean curvature hypersurfaces, Ann. Fac. Sci. Toulouse Math. (6), 18 (2009), 717-737.  doi: 10.5802/afst.1222.  Google Scholar

[24]

M. Jleli and F. Pacard, An end-to-end construction for compact constant mean curvature surfaces, Pacific J. Math., 221 (2005), 81-108.  doi: 10.2140/pjm.2005.221.81.  Google Scholar

[25]

R.V. Kohn and P. Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69-84.  doi: 10.1017/S0308210500025026.  Google Scholar

[26]

R. Mazzeo and F. Pacard, Bifurcating nodoids, In Topology and geometry: Commemorating SISTAG, volume 314 of Contemp. Math. , pages 169-186. Amer. Math. Soc. , Providence, RI, 2002. Google Scholar

[27]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.  doi: 10.1007/BF00251230.  Google Scholar

[28]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom., 64 (2003), 359-423.   Google Scholar

[29]

F. Pacard and J. Wei, Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, J. Funct. Anal., 264 (2013), 1131-1167.  doi: 10.1016/j.jfa.2012.03.010.  Google Scholar

[30]

L.A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\textbf{R}^{n}$, Arch. Rational Mech. Anal., 81 (1983), 181-197.  doi: 10.1007/BF00250651.  Google Scholar

[31]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260.  doi: 10.1007/BF00253122.  Google Scholar

[32]

J. Wei and M. Winter, On the stationary Cahn-Hilliard equation: Bubble solutions, SIAM J. Math. Anal., 29 (1998), 1492-1518 (electronic).  doi: 10.1137/S0036141097320663.  Google Scholar

[33]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.  doi: 10.1016/S0294-1449(98)80031-0.  Google Scholar

show all references

References:
[1]

N. D. AlikakosP. W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205.  doi: 10.1007/BF00375025.  Google Scholar

[2]

N. D. AlikakosX. Chen and G. Fusco, Motion of a droplet by surface tension along the boundary, Calc. Var. Partial Differential Equations, 11 (2000), 233-305.  doi: 10.1007/s005260000052.  Google Scholar

[3]

N.D. Alikakos and G. Fusco, The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions, Indiana Univ. Math. J., 42 (1993), 637-674.  doi: 10.1512/iumj.1993.42.42028.  Google Scholar

[4]

N.D. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: The motion of bubbles, Arch. Rational Mech. Anal., 141 (1998), 1-61.  doi: 10.1007/s002050050072.  Google Scholar

[5]

N.D. AlikakosG. Fusco and V. Stefanopoulos, Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, J. Differential Equations, 126 (1996), 106-167.  doi: 10.1006/jdeq.1996.0046.  Google Scholar

[6]

P.W. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations, 160 (2000), 283-356.  doi: 10.1006/jdeq.1999.3660.  Google Scholar

[7]

X. Chen, Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces, Comm. Partial Differential Equations, 19 (1994), 1371-1395.  doi: 10.1080/03605309408821057.  Google Scholar

[8]

X. Chen and M. Kowalczyk, Existence of equilibria for the Cahn-Hilliard equation via local minimizers of the perimeter, Comm. Partial Differential Equations, 21 (1996), 1207-1233.  doi: 10.1080/03605309608821223.  Google Scholar

[9]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, volume 251 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], SpringerVerlag, New York-Berlin, 1982. Google Scholar

[10]

H. DangP.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

[11]

M. del PinoM. KowalczykF. Pacard and J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516.  doi: 10.1016/j.aim.2010.01.003.  Google Scholar

[12]

M. del PinoM. Kowalczyk and J. Wei, On De Giorgi's conjecture in dimension N ≥ 9, Ann. of Math. (2)(7), 174 (2011), 1485-1569.  doi: 10.4007/annals.2011.174.3.3.  Google Scholar

[13]

M. del PinoM. Kowalczyk and J. Wei, Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in ${{\mathbb{R}}^{3}}$, Journ. Diff. Geometry, 93 (2013), 67-131.   Google Scholar

[14]

M. del PinoM. Kowalczyk and J. Wei, Traveling waves with multiple and non-convex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.  doi: 10.1002/cpa.21438.  Google Scholar

[15]

M. del PinoF. Pacard and M. Musso, Solutions of the Allen-Cahn equation which are invariant under screw-motion, Manuscripta Math., 138 (2012), 273-286.  doi: 10.1007/s00229-011-0492-3.  Google Scholar

[16]

C. Delaunay, Sur la surface de revolution dont la courbure moyenna est constante, J. Math. Pures Appl., 6 (1841), 309-320.   Google Scholar

[17]

J. Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57.  doi: 10.1007/BF03023575.  Google Scholar

[18]

P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations, 2000, pages No. 48, 26 pp. (electronic). Google Scholar

[19]

P.C. Fife, Pattern formation in gradient systems, In Handbook of dynamical systems, NorthHolland, Amsterdam, 2 (2002), 677-722.  doi: 10.1016/S1874-575X(02)80034-0.  Google Scholar

[20]

Á. Hernández and M. Kowalczyk, Delaunay end solutions of the cahn-hilliard equation in, in ´ preparation. Google Scholar

[21]

W.-y. Hsiang and W.C. Yu, A generalization of a theorem of Delaunay, J. Differential Geom., 16 (1981), 161-177.   Google Scholar

[22]

J.E. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der {W}aals-Cahn-Hilliard theory, Calc. Var. Partial Differential Equations, 10 (2000), 49-84.  doi: 10.1007/PL00013453.  Google Scholar

[23]

M. Jleli, End-to-end gluing of constant mean curvature hypersurfaces, Ann. Fac. Sci. Toulouse Math. (6), 18 (2009), 717-737.  doi: 10.5802/afst.1222.  Google Scholar

[24]

M. Jleli and F. Pacard, An end-to-end construction for compact constant mean curvature surfaces, Pacific J. Math., 221 (2005), 81-108.  doi: 10.2140/pjm.2005.221.81.  Google Scholar

[25]

R.V. Kohn and P. Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69-84.  doi: 10.1017/S0308210500025026.  Google Scholar

[26]

R. Mazzeo and F. Pacard, Bifurcating nodoids, In Topology and geometry: Commemorating SISTAG, volume 314 of Contemp. Math. , pages 169-186. Amer. Math. Soc. , Providence, RI, 2002. Google Scholar

[27]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.  doi: 10.1007/BF00251230.  Google Scholar

[28]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom., 64 (2003), 359-423.   Google Scholar

[29]

F. Pacard and J. Wei, Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, J. Funct. Anal., 264 (2013), 1131-1167.  doi: 10.1016/j.jfa.2012.03.010.  Google Scholar

[30]

L.A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\textbf{R}^{n}$, Arch. Rational Mech. Anal., 81 (1983), 181-197.  doi: 10.1007/BF00250651.  Google Scholar

[31]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260.  doi: 10.1007/BF00253122.  Google Scholar

[32]

J. Wei and M. Winter, On the stationary Cahn-Hilliard equation: Bubble solutions, SIAM J. Math. Anal., 29 (1998), 1492-1518 (electronic).  doi: 10.1137/S0036141097320663.  Google Scholar

[33]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.  doi: 10.1016/S0294-1449(98)80031-0.  Google Scholar

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