Rigidity and gamma convergence for solid‐solid phase transitions with SO(2) invariance

Abstract The singularly perturbed two‐well problem in the theory of solid‐solid phase transitions takes the form where u : Ω ⊂ ℝ n → ℝ n is the deformation, and W vanishes for all matrices in K = SO( n ) A ∪ SO( n ) B . We focus on the case n = 2 and derive, by means of Gamma convergence, a sharp‐interface limit for I ε . The proof is based on a rigidity estimate for low‐energy functions. Our rigidity argument also gives an optimal two‐well Liouville estimate: if ∇ u has a small BV norm (compared to the diameter of the domain), then, in the L 1 sense, either the distance of ∇ u from SO(2) A or the one from SO(2) B is controlled by the distance of ∇ u from K . This implies that the oscillation of ∇ u in weak L 1 is controlled by the L 1 norm of the distance of ∇ u to K . © 2006 Wiley Periodicals, Inc.