Rotationally symmetric solutions to the Cahn-Hilliard equation

Álvaro Hernández; Michał Kowalczyk

doi:10.3934/dcds.2017033

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2017, Volume 37, Issue 2: 801-827. Doi: 10.3934/dcds.2017033

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Rotationally symmetric solutions to the Cahn-Hilliard equation

Álvaro Hernández^1, and
Michał Kowalczyk^2,

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

Author Bio: E-mail address: ahernandez@dim.uchile.cl; E-mail address: kowalczy@dim.uchile.cl

Received: March 31, 2015

Revised: October 31, 2015

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary:35J61, 35B08, 35B07, 35B10, 35B36; Secondary: 35Q56, 35Q79.

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