Abstract:

The following curious problem appeared in connection with applications of calculus of variations in material science (it was probably first formulated by John Ball):

Suppose that A is a finite set of 2 x 2 matrices for which there is a non-affine deformation of the plane whose derivative belongs to A, for almost all x. Is it true that then there are matrices A,B in A such that rank(B-A)=1? (A deformation of the plane is a Lipschitz mapping of the plane into itself.)

Such deformations minimize of the integral int_Omega F(nabla u) dx, where F(A) is the distance of A from A, and so form a reasonable model for the case of `n-well potential F.' (Here n is the number of elements of A.) One may expect that the solutions have the structure of laminates; this may mean that (some subregion of) Omega can be partitioned into parallel stripes in each of which u has constant derivative. In particular, the derivatives in two such neighboring stripes have to differ by a rank one matrix.

A simple linear algebra shows that the answer to the above question is positive for n=2. The case n=3 it not so simple, but it has been answered positively a few years ago by Sverak. Very recently, the problem has been completely solved: For n=4 the answer is still positive (Chlebik and Kirchheim), but for n=5 the answer is negative (Kirchheim and Preiss). Without going to technical details, the talk will present ideas (of the type of Gromov's `convex integration') used to transform the construction for n=5 to a simple geometric question in the three dimensional space and try to explain the picture that gives the solution.