  We establish a quantitative rigidity estimate for two-well frame-indifferent
nonlinear energies, in the case in which the two wells have exactly one
rank-one connection. Building upon this novel rigidity result, we then analyze
solid-solid phase transitions in arbitrary space dimensions, under a suitable
anisotropic penalization of second variations. By means of
$\Gamma$-convergence, we show that, as the size of transition layers tends to
zero, singularly perturbed two-well problems approach an effective
sharp-interface model. The limiting energy is finite only for deformations
which have the structure of a laminate. In this case, it is proportional to the
total length of the interfaces between the two phases.
