Past Applied Analysis and Mechanics Seminar

24 May 2004
17:00
Stefan Wenger
Abstract
Integral currents were introduced by H. Federer and W. H. Fleming in 1960 as a suitable generalization of surfaces in connection with the study of area minimization problems in Euclidean space. L. Ambrosio and B. Kirchheim have recently extended the theory of currents to arbitrary metric spaces. The new theory provides a suitable framework to formulate and study area minimization and isoperimetric problems in metric spaces. The aim of the talk is to discuss such problems for Banach spaces and for spaces with an upper curvature bound in the sense of Alexandrov. We present some techniques which lead to isoperimetric inequalities, solutions to Plateau's problem, and to other results such as the equivalence of flat and weak convergence for integral currents.
  • Applied Analysis and Mechanics Seminar
17 May 2004
17:00
Bryan Rynne
Abstract
We consider semilinear Sturm-Liouville and elliptic problems with jumping nonlinearities. We show how `half-eigenvalues' can be used to describe the solvability of such problems and consider the structure of the set of half-eigenvalues. It will be seen that for Sturm-Liouville problems the structure of this set can be considerably more complicated for periodic than for separated boundary conditions, while for elliptic partial differential operators only partial results are known about the structure in general.
  • Applied Analysis and Mechanics Seminar
8 March 2004
17:00
Friedemann Schuricht
Abstract
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.
  • Applied Analysis and Mechanics Seminar
1 March 2004
17:00
Guiseppe Mingione
Abstract
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.
  • Applied Analysis and Mechanics Seminar
23 February 2004
17:00
Carsten Carstensen
Abstract
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 <em>not</em> attained. In the sense of (Sobolev) functions, the non-rank-one convex minimisation problem (<em>M</em>) 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 (<em>R</em>), 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.
  • Applied Analysis and Mechanics Seminar

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