Gamma-Limit for the Extended Fisher-Kolmogorov equation

We consider the Extended Fisher-Kolmogorov equation
${\partial }_{i}u+{ϵ}^2\gamma {\mathrm{\Delta }}^{2}u-\mathrm{\Delta }u+\frac{1}{{ϵ}^2}F‘(u)=0$
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 $\mathrm{\Gamma }$;-Limit of
${ϵ}_{ϵ}^{\gamma }(u):=\underset{\mathrm{\Omega }}{\int }\frac{{ϵ}^3\gamma }{2}|\mathrm{\Delta }u{|}^2+\frac{ϵ}{2}|\mathrm{\nabla }u{|}^2+\frac{1}{ϵ}F(u)$ which is a Ljapunov functional of (1).
This is joint work with Danielle Hilhorst and Lambert A. Peletier.