Rigidity and Gamma convergence for solid-solid phase transitions with SO(2)-invariance
Sergio Conti and Ben Schweizer
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Submission date: 12. Oct. 2004
published in: Communications on pure and applied mathematics, 59 (2006) 6, p. 830-868
DOI number (of the published article): 10.1002/cpa.20115
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The singularly perturbed two-well problem in the theory of solid-solid phase transitions takes the form
where is the deformation, and W vanishes for all matrices in . We focus on the case n=2 and derive, by means of Gamma convergence, a sharp-interface limit for . The proof is based on a rigidity estimate for low-energy functions. Our rigidity argument also gives an optimal two-well Liouville estimate: if has a small BV norm (compared to the diameter of the domain), then, in the sense, either the distance of from SO(2)A or the one from SO(2)B is controlled by the distance of from K. This implies that the oscillation of in weak- is controlled by the norm of the distance of to K.