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The singularly perturbed two‐well problem in the theory of solid‐solid phase transitions takes the form where u : Ω ⊂ ℝⁿ → ℝⁿ 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 L¹ 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 L¹ is controlled by the L¹ norm of the distance of ∇u to K. © 2006 Wiley Periodicals, Inc.