Past Seminars › Seminar Series

Applied Analysis and Mechanics Seminar (past)

Mon, 24/05/2004
17:00 		Stefan Wenger (ETH-Zurich) Applied Analysis and Mechanics Seminar Add to calendar L1
Currents in metric spaces, isoperimetric inequalities, and applications to area minimization problems
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.
Mon, 17/05/2004
17:00 		Bryan Rynne (Heriot-Watt) Applied Analysis and Mechanics Seminar Add to calendar L1
Half-eigenvalues and semilinear problems with jumping nonlinearities
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.
Mon, 26/04/2004
17:00 		Scott Spector (Southern Illinois University) Applied Analysis and Mechanics Seminar Add to calendar L1
Cavitation and Configurational Forces in a Nonlinearly Elastic Material
Mon, 08/03/2004
17:00 		Friedemann Schuricht (Cologne) Applied Analysis and Mechanics Seminar Add to calendar L1
Ideal Knots
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.
Mon, 01/03/2004
17:00 		Guiseppe Mingione (Parma) Applied Analysis and Mechanics Seminar Add to calendar L1
Elliptic systems, integral functionals and singular sets
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.
Mon, 23/02/2004
17:00 		Carsten Carstensen (Bristol) Applied Analysis and Mechanics Seminar Add to calendar L1
Adaptive finite elements for relaxed methods (FERM) in computational microstructures
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.
Mon, 09/02/2004
17:00 		John Norbury (Oxford) Applied Analysis and Mechanics Seminar Add to calendar L1
Perturbed Ginzberg-Landau systems
Mon, 02/02/2004
17:00 		Daniel Faraco (Max Planck Leipzig) Applied Analysis and Mechanics Seminar Add to calendar L1
Geometric rigidity of conformal matrices
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.
Mon, 26/01/2004
17:00 		Jonathan Bevan (Oxford) Applied Analysis and Mechanics Seminar Add to calendar L1
Polyconvexity and counterexamples to regularity in the calculus of variations
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.
Wed, 07/01/2004
16:00 		N Kikuchi (Keio University) Applied Analysis and Mechanics Seminar Add to calendar SR1
On the regularity of discrete Morse flows
Mon, 01/12/2003
17:00 		Jan Maly (Prague) Applied Analysis and Mechanics Seminar Add to calendar L1
Jacobians and weak convergence
Mon, 17/11/2003
17:00 		Dr Andrew Lorent (Oxford) Applied Analysis and Mechanics Seminar Add to calendar L1
The Aviles Giga functional
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.
Mon, 20/10/2003
17:00 		Jose Carlos Bellido (Universidad de Castilla-La Mancha/Oxford) Applied Analysis and Mechanics Seminar Add to calendar L1
A variational approach to optimal design

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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