Austenite as a Local Minimizer in a Model of Material Microstructure with a Surface Energy Term

Abstract. A family of integral functionals F which model in a simplified way material microstructure occupying a two-dimensional domain Ω and which take account of surface energy and a variable well depth is studied. It is shown that there is a critical well depth, whose scaling with the surface energy density and domain dimensions is given, below which the state u = 0 is the global minimizer of a typical F in F . It is also shown that u = 0 is a strict local minimizer of F in the sense that if v = 0 is admissible and either ||v||L2(Ω) or L2({(x, y) ∈ Ω : |vy|(x, y) ≥ 1}) is sufficiently small (with quantitative bounds given in terms of the parameters appearing in the energy functional F ), then F (v) > F (0). Provided the well depth is sufficiently large, the existence of a so-called energy barrier between u = 0 and the global minimizer of F is established under the assumption that paths (v(t))0≤t≤1 connecting these two states obey |vy| ≤ 1 almost everywhere in the domain.