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

Abstract

The second-order singularly-perturbed problem concerns the integral functional ${\int }_{\Omega }{\epsilon }_n^{-1}W(u)+{\epsilon }_n^3\Vert {\nabla }^{2}u{\Vert }^{2}dx$ for a bounded open set $\Omega \subseteq {\mathbb{R}}^N$, a sequence ${\epsilon }_n\to 0^{+}$ 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.

37 pages