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

The second-order singularly-perturbed problem concerns the integral functional ${\int }_{\mathrm{\Omega }}{\epsilon }_n^{-1}W(u)+{\epsilon }_n^3\mathrm{\parallel }{\mathrm{\nabla }}^{2}u{\mathrm{\parallel }}^{2}dx$ for a bounded open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$, a sequence ${\epsilon }_n\to 0^{+}$ of positive reals, and a function $W:\mathbb{R}\to [0,\mathrm{\infty })$ with exactly two distinct zeroes. This functional is of interest since it models the behavior of phase transitions, and its Gamma limit as $n\to \mathrm{\infty }$ 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.