A lower bound for a variational model for pattern formation in shape-memory alloys

The Kohn-Müller model for the formation of domain patterns in martensitic shape-memory alloys consists in minimizing the sum of elastic, surface and boundary energy in a simplified scalar setting, with a nonconvex constraint representing the presence of different variants. Precisely, one minimizes among all u:(0,l)×(0,h)→ ℝ such that ∂ _y u = ± 1 almost everywhere. We prove that for small ε the minimum of J _{ε, β} scales as the smaller of ε^1/2β^1/2 l ^1/2 h and ε^2/3 l ^1/3 h, as was conjectured by Kohn and Müller. Together with their upper bound, this shows rigorously that a transition is present between a laminar regime at ε/l≫ β³ and a branching regime at ε/l≪ β³. PACS 64.70.Kb, 62.20.-x, 02.30.Xx