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Talk

Gamma-Limit for the Extended Fisher-Kolmogorov equation

  • Reiner Schaetzle (Universität Freiburg)
A3 01 (Sophus-Lie room)

Abstract

We consider the Extended Fisher-Kolmogorov equation
$\partial_i u +\epsilon^2 \gamma \Delta^2 u - \Delta u + \frac{1}{\epsilon^2} F` (u) =0 $
where $F(t):= \frac{1}{4} (t^2 -1 )^2$ is a double-well potential and $\gamma >0$.For $\gamma =0$ this is the ordinary Allen-Cahn equation. The equation for the stationary waves is the ordinary differential equation
$\gamma U```` - U`` +F` (U) =0$
with appropriate boundary conditions. In this talk, we present estimates on the second derivatives of solutions of (2) which enable to prove that bumps of these solutions have to have a minimal size, hence cannot accumulate.
As main result, we prove that the area functional is the $\Gamma$;-Limit of
$\epsilon^\gamma_\epsilon (u):=\int\limits_{\Omega} \frac{\epsilon^3 \gamma}{2} |\Delta u|^2 + \frac{\epsilon}{2} |\nabla u |^2 + \frac{1}{\epsilon} F(u)$ which is a Ljapunov functional of (1).
This is joint work with Danielle Hilhorst and Lambert A. Peletier.

seminar
26.11.96 30.01.25

Oberseminar Analysis

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E2 10 (Leon-Lichtenstein) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Augusteum - A314