Γ-convergence of higher-order phase transition models

We investigate the asymptotic behavior as $\varepsilon \to 0$ of singularly perturbed phase transition models of order $n\ge 2$, given by $G_{\varepsilon }^{\lambda ,n}\left[u\right]≔{\int }_I\frac{1}{\varepsilon }W\left(u\right)-\lambda {\varepsilon }^{2n-3}{\left(u^{(n-1)}\right)}^2+{\varepsilon }^{2n-1}{\left(u^{\left(n\right)}\right)}^{2}dx,u\in W^{n,2}\left(I\right),$where $\lambda >0$ is fixed, $I\subset R$ is an open bounded interval, and $W\in C^0\left(R\right)$ is a suitable double-well potential. We find that there exists a positive critical parameter depending on $W$ and $n$, such that the $\Gamma$-limit of $G_{\varepsilon }^{\lambda ,n}$ with respect to the $L^1$-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo–Nirenberg type inequalities.