Local minimizers and low energy paths in a model of material microstructure with a surface energy term ∗

A family of integral functionals F which, in a simplified way, model 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|| L 2 (Ω) or L 2 ({(x, y) ∈ Ω : |v y |(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). Low energy paths between u = 0 and the global minimizer (in the case of a sufficiently large well-depth) are given such that the cost of introducing sets {(x, y) ∈ Ω : |v y (x, y)| ≥ 1} of positive measure into the domain Ω may be made arbitrarily small.