L1

Nonconvex minimisation problems are encountered in many applications such as phase transitions in solids (1) or liquids but also in optimal design tasks (2) or micromagnetism (3). In contrast to rubber-type elastic materials and many other variational problems in continuum mechanics, the minimal energy may be not attained. In the sense of (Sobolev) functions, the non-rank-one convex minimisation problem (M) is ill-posed: As illustrated in the introduction of FERM, the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures. Some macroscopic or effective quantities, however, are well-posed and the target of an efficient numerical treatment. The presentation proposes adaptive mesh-refining algorithms for the finite element method for the effective equations (R), i.e. the macroscopic problem obtained from relaxation theory. For some class of convexified model problems, a~priori and a~posteriori error control is available with an reliability-efficiency gap. Nevertheless, convergence of some adaptive finite element schemes is guaranteed. Applications involve model situations for (1), (2), and (3) where the relaxation is provided by a simple convexification.

Recently Friesecke, James and Muller established the following

quantitative version of the rigidity of SO(n) the group of special orthogonal

matrices. Let U be a bounded Lipschitz domain. Then there exists a constant

C(U) such that for any mapping v in the L2-Sobelev space the L^2-distance of

the gradient controlls the distance of v a a single roation.

This interesting inequality is fundamental in several problems concerning

dimension reduction in nonlinear elasticity.

In this talk, we will present a joint work with Muller and Zhong where we

investigate an analagous quantitative estimate where we replace SO(n) by an

arbitrary smooth, compact and SO(n) invariant subset of the conformal

matrices E. The main novelty is that exact solutions to the differential

inclusion Df(x) in E a.e.x in U are not necessarily affine mappings.

Using a technique explored in unpublished work of Ball and Mizel I shall

show that already in 2 and 3 dimensions there are vectorfields which are

singular minimizers of integral functionals whose integrand is strictly

polyconvex and depends on the gradient of the map only. The analysis behind

these results gives rise to an interesting question about the relationship

between the regularity of a polyconvex function and that of its possible

convex representatives. I shall indicate why this question is interesting in

the context of the regularity results above and I shall answer it in certain

cases.