Gamma-Limit for the Extended Fisher-Kolmogorov equation
- Reiner Schaetzle (Universität Freiburg)
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.