Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Γ-limit for the extended Fisher–Kolmogorov equation

D. Hilhorsta1, L. A. Peletiera2 and R. Schätzlea3 p1

a1 Analyse Numérique et EDP, CNRS, et Université de Paris-Sud, 91405 Orsay, France

a2 Mathematical Institute, Leiden University, The Netherlands

a3 Departement der Mathematik, ETH Zentrum, Zürich, Switzerland

Abstract

We consider the Lyapunov functional,

S0308210500001566disp001

of the rescaled Extended Fisher-Kolmogorov equation
S0308210500001566disp002

This is a fourth order generalization of the Fisher–Kolmogorov or Allen–Cahn equation. We prove that if ε → 0, then S0308210500001566inline001 tends to the area functional in the sense of Γ-limits, where the transition energy is given by the one-dimensional kink of the Extended Fisher–Kolmogorov equation.

(Received May 18 2000)

(Accepted March 15 2001)

Correspondence:

p1 Mathematisches Institut der Rheinischen Friedrich-Wilhelms-Universität Bonn, Beringstraße 6, D-53115 Bonn, Germany.