Γ-Convergence of Higher Order Phase Transition Models.Oberseminar Analysis

by Gianna Götzmann (Uni Augsburg)

Wednesday Apr 1, 2026, 2:15 PM → 4:15 PM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

We investigate the asymptotic behavior as $\varepsilon \to 0$ of singularly perturbed phase transition models of order $n \geq 2$, given by
$$
G_\varepsilon^{\lambda,n}[u] :=
\int_I \frac 1\varepsilon W(u) -\lambda\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \mathcal{d} x, \quad u \in W^{n,2}(I),
$$
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 $\Gamma$-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.