Applied Analysis and Mechanics Seminar
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Mon, 26/01/2004 17:00 |
Jonathan Bevan (Oxford) |
Applied Analysis and Mechanics Seminar  |
L1 |
| Using a technique explored in unpublished work of Ball and Mizel I shall show that already in 2 and 3 dimensions there are vectorfields which are singular minimizers of integral functionals whose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. | |||
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Mon, 02/02/2004 17:00 |
Daniel Faraco (Max Planck Leipzig) |
Applied Analysis and Mechanics Seminar |
L1 |
| 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. | |||
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Mon, 09/02/2004 17:00 |
John Norbury (Oxford) |
Applied Analysis and Mechanics Seminar |
L1 |
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Mon, 16/02/2004 17:00 |
Doina Cioranescu (Paris) |
Applied Analysis and Mechanics Seminar |
L1 |
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Mon, 23/02/2004 17:00 |
Carsten Carstensen (Bristol) |
Applied Analysis and Mechanics Seminar |
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. | |||
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Mon, 01/03/2004 17:00 |
Guiseppe Mingione (Parma) |
Applied Analysis and Mechanics Seminar |
L1 |
| 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. | |||
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Mon, 08/03/2004 17:00 |
Friedemann Schuricht (Cologne) |
Applied Analysis and Mechanics Seminar |
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. | |||
