Professor David Preiss
(University College, London)
"Deformations with finitely many gradients"
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.