Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions - Conti, S; Dolzmann, G; Kirchheim, B (2007)
The three-well problem consists in looking for minimizers u : Ω ⊂ R3 → R3 of a functional I (u) = ∫Ω W (∇ u) d x, where the elastic energy W models the tetragonal phase of a phase-transforming material. In particular, W attains its minimum on K = n-ary unioni = 13 SO (3) Ui, with Ui being the three distinct diagonal matrices with eigenvalues (λ, λ, over(λ, ̃)), λ, over(λ, ̃) > 0 and λ ≠ over(λ, ̃). We show that, for boundary values F in a suitable relatively open subset of M3 × 3 ∩ {F : det F = det U1}, the differential inclusion{(∇ u ∈ K, in Ω,; u (x) = F x, on ∂ Ω) has Lipschitz solutions. © 2006 Elsevier Masson SAS. All rights reserved.