L1

Let gamma be a closed knotted curve in R^3 such that the tubular

neighborhood U_r (gamma) with given radius r>0 does not intersect

itself. The length minimizing curve gamma_0 within a prescribed knot class is

called ideal knot. We use a special representation of curves and tools from

nonsmooth analysis to derive a characterization of ideal knots. Analogous

methods can be used for the treatment of self contact of elastic rods.

I shall give a brief overview of the partial regularity results for minima

of integral functionals and solutions to elliptic systems, concentrating my

attention on possible estimates for the Hausdorff dimension of the singular

sets; I shall also include more general variational objects called almost

minimizers or omega-minima. Open questions will be discussed at the end.

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.