Qualitative Behavior of Local Minimizers of Singular Perturbed Variational Problems

Abstract

We consider a non-convex variational problem (P) and the corresponding singular perturbed problem (P ε ). The qualitative behavior of stable critical points of (P ε ) depending on ε and a lower order term is discussed and we prove compactness of a sequence of stable critical points as ε 0. Moreover we show whether this limit is the global minimizer of (P). Furthermore uniform convergence is considered as well as the convergence rate depending on ε.

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

“…Alternatively, (1.3) occurs as a quasi-steady, phase field model in the theory of solidification with ϑ representing temperature [1,2]. One-dimensional phase transition modelling can also give rise to higher order non-convex variational problems [10,12], see [9] for a general discussion. However we confine ourselves to studying generalizations of (1.2) and the weak* limits in L ∞ per of its minimizers as → 0 and two questions arise.…”

“…For example, (1.6) may have a unique minimizer. This happens if the set of zeros of ϑ is discrete, a case that was thoroughly investigated in [1,2] (see also [10] for generalizations and further discussion). Alternatively, (1.6) may have an infinite set of minimizers, including Young measure solutions, as happens when ϑ in (1.1) vanishes on some interval.…”