

Preprint 50/1998
Relaxation of some multi-well problems
Kaushik Bhattacharya and Georg Dolzmann
Contact the author: Please use for correspondence this email.
Submission date: 15. Nov. 1998
Pages: 39
published in: Proceedings of the Royal Society of Edinburgh / A, 131 (2001) 2, p. 279-320
DOI number (of the published article): 10.1017/S0308210500000883
Bibtex
MSC-Numbers: 49J40, 52A30, 73B99, 73C50, 73V25
Keywords and phrases: nonconvex variational problems, generalized convex hulls, existence of minimizers, in-approximation, relaxed energy
Download full preprint: PDF (708 kB), PS ziped (394 kB)
Abstract:
Mathematical models of phase transitions in solids lead to the variational problem, minimize where W has a multi-well structure: W = 0 on a multi-well set K and W>0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e., for
or
for
with
, in three dimensions when the matrices
are essentially two-dimensional and also for
for
with
which arises in the study of thin films. Here
denotes the
-matrix formed with the first two columns of
. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasiconvex hull to propose `approximate relaxed energies', quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.