

Preprint 69/2004
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.