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Thursday, 23th November 2000, 4:00 pm

Mathematics Bldg., lecture hall 2

Professor David Preiss

(University College, London)

"Deformations with finitely many gradients"

** 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.

Coffee, Cookies at the faculty lounge at 3:30.

You are invited to join the speaker for further discussion after the talk at Beit Belgia.

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