L1

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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.