Rigidity and Gamma convergence for solid-solid phase transitions with SO(2)-invariance

Sergio Conti and Ben Schweizer

Contact the author: Please use for correspondence this email.
Submission date: 12. Oct. 2004
Pages: 45
published in: Communications on pure and applied mathematics, 59 (2006) 6, p. 830-868 
DOI number (of the published article): 10.1002/cpa.20115
Bibtex
Download full preprint: PDF (461 kB), PS ziped (363 kB)

Abstract:
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.