Gamma-Convergence of Higher-Order Phase Transition Models

We investigate the asymptotic behavior as $\epsilon \to 0$ of singularly perturbed phase transition models of order $n\ge 2$, given by \begin{align} G_\varepsilon^{λ,n}[u] := \int_I \frac 1\varepsilon W(u) -λ\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \ dx, \quad u \in W^{n,2}(I), \end{align} where $\lambda >0$ is fixed, $I\subset \mathbb{R}$ is an open bounded interval, and $W\in C^0(\mathbb{R})$ is a suitable double-well potential. We find that there exists a positive critical parameter depending on $W$ and $n$, such that the $\mathrm{\Gamma }$-limit of $G_{\epsilon }^{\lambda ,n}$ 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.