Numerical approximation of the solution of a variational problem with a double well potential. (English) Zbl 0725.65067

Collins, Charles; Kinderlehrer, David; Luskin, Mitchell

SIAM J. Numer. Anal. 28, No. 2, 321-332 (1991).

The authors consider variational problems with a double well potential. Such problems arise in the description of equilibria of crystals or other ordered states but they are not lower semicontinuous and can fail to attain a minimum value. Functionals modelling these materials are not lower semicontinuous and a deformation attaining minimum energy does not exist for many boundary displacements.
The concept of Young measure permits to give a description of the solution. Approximation methods are studied and the authors assert that “they give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals”. From my point of view this paper is very hard to read and I am not sure that it could be used by engineers working on these subjects. No concrete application is given and the numerical method is theoretically described but not very easy to understand.

Reviewer: Y.Cherruault (Paris)

MSC:

65K10	Numerical optimization and variational techniques
82D25	Statistical mechanics of crystals
49K20	Optimality conditions for problems involving partial differential equations
49M15	Newton-type methods

Keywords:

variational problems; double well potential; Young measure; solid crystals