A sharp-interface limit for a two-well problem in geometrically linear elasticity Sergio Conti 1 and Ben Schweizer 2 April 21, 2005 Abstract: In the theory of solid-solid phase transitions the deformation u :Ω ⊂ R d → R d of an elastic body is determined via a functional containing a nonconvex energy density and a singular perturbation. We study I ε [u]=  Ω 1 ε W (∇u)+ ε|∇ 2 u| 2 . Frame indifference, within a linearized setting, requires that W depends only on the symmetric part of ∇u. The potential W is non-negative and vanishes on two wells, i.e., for d = 2, on two lines in the space of matrices. We determine, for d = 2, the Gamma limit I 0 =Γ − lim ε→0 I ε . The limit I 0 [u] is finite only for deformations u that fulfill W (∇u) = 0 almost everywhere and have sharp interfaces where ∇u has jumps. For these u, I 0 [u] equals the line in- tegral over the interfaces of a surface energy density. 1 Introduction Modeling of phase transitions in solids leads to functionals of the form E ε [u, Ω] =  Ω W (∇u)+ ε 2 |∇ 2 u| 2 , (1.1) where u : R d ⊃ Ω → R d is the elastic displacement, W a free energy density with multiple minima, and ε> 0 a small parameter. The free energy prefers deformations u with ∇u near the set of minima of W , the second part of the energy penalizes transitions from one gradient to another. The parameter ε determines the width of domain walls [5, 19, 24, 7]. Variational problems as in (1.1) have often been proposed both for numer- ical and analytical computations, but the presence of different length scales 1 Fachbereich Mathematik, Universit¨ at Duisburg-Essen, Campus Duisburg, Lotharstr. 65, D-47057 Duisburg, Germany 2 Mathematisches Institut, Universit¨ at Basel, Rheinsprung 21, CH-4051 Basel, Schweiz 1