Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions

The three-well problem consists in looking for minimizers u: Omega subset of R-3 -> R-3 of a functional I(u) = integral del W(del u)dx, where the elastic energy W models the tetragonal phase of a phase-transforming material. In particular, W attains its minimum on K = boolean OR(3)(i=1) SO(3)Ui, with Ui being the three distinct diagonal matrices with eigenvalues (lambda, lambda,(lambda) over bar), lambda, lambda > 0 and lambda not equal (lambda) over bar. We show that, for boundary values F in a suitable relatively open subset of M-3 x3 boolean AND {F: det F = det U-1}, the differential inclusion [GRAPHICS] has Lipschitz solutions. (c) 2006 Elsevier Masson SAS. All rights reserved.