Effect of Boundary Conditions on Second-Order Singularly-Perturbed Phase Transition Models on $\mathbb{R}$

TL;DR

This paper investigates how boundary conditions influence the Gamma limit of a second-order singularly-perturbed phase transition model in one dimension, revealing a different limit form when boundary data is included.

Contribution

It introduces a new form of the Gamma limit for the model in one dimension and analyzes the impact of boundary conditions on this limit.

Findings

Different Gamma limit form identified with boundary conditions

Boundary data significantly affects the phase transition behavior

Results extend understanding of boundary effects in singular perturbation problems

Abstract

The second-order singularly-perturbed problem concerns the integral functional $\int_\Omega \varepsilon_n^{-1}W(u) + \varepsilon_n^3\|\nabla^2u\|^2\,dx$ for a bounded open set $\Omega \subseteq \mathbb{R}^N$, a sequence $\varepsilon_n \to 0^+$ of positive reals, and a function $W:\mathbb{R} \to [0,\infty)$ with exactly two distinct zeroes. This functional is of interest since it models the behavior of phase transitions, and its Gamma limit as $n \to \infty$ was studied by Fonseca and Mantegazza. In this paper, we study an instance of the problem for $N=1$. We find a different form for the Gamma limit, and study the Gamma limit under the addition of boundary data.