Many different physical systems, e.g. super-coiled DNA molecules, have been successfully modelled as elastic curves, ribbons or rods. We will describe all such systems as framed curves, and will consider problems in which a three dimensional framed curve has an associated energy that is to be minimized subject to the constraint of there being no selfintersection. For closed curves the knot type may therefore be specified a priori. Depending on the precise form of the energy and imposed boundary conditions, local minima of both open and closed framed curves often appear to involve regions of self-contact, that is, regions in which points that are distant along the curve are close in space. While this phenomenon of self-contact is familiar through every day experience with string, rope and wire, the idea is surprisingly difficult to define in a way that is simultaneously physically reasonable, mathematically precise, and analytically tractable. Here we use the notion of global radius of curvature of a space curve in a new formulation of the self-contact constraint, and exploit our formulation to derive existence results for minimizers, in the presence of self-contact, of a range of elastic energies that define various framed curve models. As a special case we establish the existence of ideal shapes of knots. (2000): 49J99, 53A04, 57M25, 74B20, 92C40 Mathematics Subject Classification

We consider a number of problems that are associated with the 1-Laplace operator Div (Du/|Du|), the formal limit of the p-Laplace operator for p → 1, by investigating the underlying variational problem. Since corresponding solutions typically belong to BV and not to W 1,1 , we have to study minimizers of functionals containing the total variation. In particular we look for constrained minimizers subject to a prescribed L 1 -norm which can be considered as an eigenvalue problem for the 1-Laplace operator. These variational problems are neither smooth nor convex. We discuss the meaning of Dirichlet boundary conditions and prove existence of minimizers. The lack of smoothness, both of the functional to be minimized and the side constraint, requires special care in the derivation of the associated Euler-Lagrange equation as necessary condition for minimizers. Here the degenerate expression Du/|Du| has to be replaced with a suitable vector field z ∈ L ∞ to give meaning to the highly singular 1-Laplace operator. For minimizers of a large class of problems containing the eigenvalue problem, we obtain the surprising and remarkable fact that in general infinitely many Euler-Lagrange equations have to be satisfied.

The investigation of contact interactions, such as traction and heat flux, that are exerted by contiguous bodies across the common boundary is a fundamental issue in continuum physics. However, the traditional theory of stress established by Cauchy and extended by Noll and his successors is insufficient for handling the lack of regularity in continuum physics due to shocks, corner singularities, and fracture. This paper provides a new mathematical foundation for the treatment of contact interactions. Based on mild physically motivated postulates, which differ essentially from those used before, the existence of a corresponding interaction tensor is established. While in earlier treatments contact interactions are basically defined on surfaces, here contact interactions are rigorously considered as maps on pairs of subbodies. This allows the action exerted on a subbody to be defined not only, as usual, for sets with a sufficiently regular boundary, but also for Borel sets (which include all open and all closed sets). In addition to the classical representation of such interactions by means of integrals on smooth surfaces, a general representation using the distributional divergence of the tensor is derived. In the case where concentrations occur, this new approach allows a more precise description of contact phenomena than before.

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We present a characterization of ideal knots, i.e., of closed knotted curves of prescribed thickness with minimal length, where we use the notion of global curvature for the definition of thickness. We show with variational methods that for an ideal knot γ, the normal vector γ (s) at a curve point γ(s) is given by the integral over all vectors γ(τ )−γ(s) against a Radon measure, where |γ(τ ) − γ(s)|/2 realizes the given thickness. As geometric consequences we obtain in particular, that points without contact lie on straight segments of γ, and for points γ(s) with exactly one contact point γ(τ ) we have that γ (s) points exactly into the direction of γ(τ ) − γ(s). Moreover, isolated contact points lie on straight segments of γ, and curved arcs of γ consist of contact points only, all realizing the prescribed thickness with constant (maximal) global curvature. (2000): 53A04, 57M25, 74K05, 74M15, 92C40 Mathematics Subject Classification

We study in detail the notion of global curvature defined on rectifiable closed curves, a concept which has been successfully applied in existence and regularity investigations regarding elastic self-contact problems in nonlinear elasticity. A bound on this purely geometric quantity serves as an excluded volume constraint to prevent selfintersections of slender elastic bodies modeled as elastic rods. Moreover, a finite global curvature characterizes simple closed curves, whose arc length parameterizations possess a Lipschitz continuous tangent field. The investigation of local and non-local properties of global curvature motivates, in particular, an extended definition of local curvature at any point of a rectifiable loop. Finally we show how a bound on global curvature can be used to define and control topological constraints such as a given knot type for closed loops or a prescribed linking number for closed framed curves, suitable to describe, e.g., supercoiling phenomena of biomolecules. (2000): 53A04, 57M25, 74K10, 74M15, 92C40 Mathematics Subject Classification

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