Past Applied Analysis and Mechanics Seminar

2 February 2004
17:00
Daniel Faraco
Abstract
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.
  • Applied Analysis and Mechanics Seminar
26 January 2004
17:00
Jonathan Bevan
Abstract
Using a technique explored in unpublished work of Ball and Mizel I shall show that already in 2 and 3 dimensions there are vectorfields which are singular minimizers of integral functionals whose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.
  • Applied Analysis and Mechanics Seminar
17 November 2003
17:00
Dr Andrew Lorent
Abstract
Take any region omega and let function u defined inside omega be the distance from the boundary, u solves the iconal equation \lt|Du\rt|=1 with boundary condition zero. Functional u is also conjectured (in some cases proved) to be the "limiting minimiser" of various functionals that arise models of blistering and micro magnetics. The precise formulation of these problems involves the notion of gamma convergence. The Aviles Giga functional is a natural "second order" generalisation of the Cahn Hilliard model which was one of the early success of the theory of gamma convergence. These problems turn out to be surprisingly rich with connections to a number of areas of pdes. We will survey some of the more elementary results, describe in detail of one main problems in field and state some partial results.
  • Applied Analysis and Mechanics Seminar

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