Phase transitions with a minimal number of jumps in the singular limits of higher order theories

Abstract

For a smooth $W:(0,\infty )\times {\mathbb{R}}^d\to \mathbb{R}$ and a family of $L$-periodic $W^{1,2}$-functions ${\vartheta }_{ϵ}:\mathbb{R}\to {\mathbb{R}}^d$ with ${\vartheta }_{ϵ}\rightharpoonup \vartheta$, the basic problem is to understand the weak* limit as $ϵ\to 0$ of $L$-periodic minimizers of

It is assumed that $W(\varphi ,\theta )\to \infty$ as $\varphi \to 0,\infty$, and that $W(\cdot ,\theta )$, which has no more than three critical points counting multiplicity depending on $\theta \in {\mathbb{R}}^d$, is of a type that arises in the Cahn–Hilliard theory of phase separations where $d=1$. The limiting problem with $ϵ=0$ is to minimize, over bounded $L$-periodic measurable functions $\phi$,

Minimizers of (‡) need not be unique (there may be uncountably many), they may be discontinuous and minimizers with only simple jumps may coexist with minimizers with much more complicated discontinuities. Weak* limits of minimizers of (†) as $ϵ\to 0$ are minimizers of the relaxation of (‡). However it is shown that if, for a sequence of minimizers of (†),

then the weak* limit of any subsequence of $\{{\phi }_{{ϵ}_k}\}$ is an actual minimizer of (‡) which is continuous except at a finite number of simple jumps. Moreover, for sequences ${ϵ}_k\to 0$ from a set of positive Lebesgue density, it is shown that the weak* limit of $L$-periodic minimizers of (†) is a minimizer of (‡) with a finite number of simple jumps. Under additional hypotheses it is shown that, for sequences from a set of full Lebesgue density, the weak* limits of $L$-periodic minimizers of (†) are minimizers of (‡) with a minimal number of simple jumps.