Gamma convergence for phase transitions in impenetrable elastic materials 1 Gamma convergence for phase transitions in impenetrable elastic materials Sergio Conti and Ben Schweizer Abstract: We study the family of functionals I ε [u]=  Ω 1 ε W (∇u)+ ε|∇ 2 u| 2 dx, with u :Ω ⊂ R n → R n representing the deformation of an elastic body, and W the energy density, which vanishes for all matrices in K = SO(n)A ∪ SO(n)B. The energy I ε describes an elastic material with two preferred gradients and surface tension, the so-called two-well problem of solid-solid phase transitions. The Gamma limit of the functionals I ε was determined, for n = 2, in [5], the crucial step in the proof is to derive rigidity estimates in order to control the local rotations of minimizing sequences. While [5] treats the case that W has quadratic growth at infinity, we treat here the case that W does not permit self-penetration, i.e. W (F )= ∞ for det F< 0. We restrict to n = 2 and exploit results of [5]. 1. Introduction. Our investigations follow in spirit the pioneering work of Modica and Mortola who studied in [9] functionals of the type J ε [v]=  Ω 1 ε W (v)+ ε|∇v| 2 dx, (1) with W ≥ 0, W (ξ ) = 0 iff ξ ∈{a, b}. They found that sequences v ε with bounded energies J ε [v ε ] are precompact and that limits have a particular form, namely u 0 (x)= aχ E +b(1 −χ E ), where χ E is the characteristic function of a set E with bounded perimeter. Moreover, the (minimal) limiting energy of sequences u ε → u 0 can be characterized and is proportional to the perimeter of E. The precise statement is that the Gamma limit of the functionals J ε is