Preprint 69/2004

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
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
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Abstract:
The singularly perturbed two-well problem in the theory of solid-solid phase transitions takes the form
displaymath9
where formula11 is the deformation, and W vanishes for all matrices in formula15. We focus on the case n=2 and derive, by means of Gamma convergence, a sharp-interface limit for formula19. The proof is based on a rigidity estimate for low-energy functions. Our rigidity argument also gives an optimal two-well Liouville estimate: if formula21 has a small BV norm (compared to the diameter of the domain), then, in the formula25 sense, either the distance of formula21 from SO(2)A or the one from SO(2)B is controlled by the distance of formula21 from K. This implies that the oscillation of formula21 in weak-formula25 is controlled by the formula25 norm of the distance of formula21 to K.

03.07.2017, 01:41