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Forthcoming events in this series
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Convexity on Grassmannians and calculus of variations
Abstract
The talk will discuss the variationnal problem on finite
dimensional normed spaces and Finsler manifolds.
We first review different notions of ellipticity (convexity) for
parametric integrands (densities) on normed spaces and compare them with
different minimality properties of affine subspaces. Special attention will
be given to Busemann and Holmes-Thompson k-area. If time permits, we will
then present the first variation formula on Finsler manifolds and exhibit a
class of minimal submanifolds.
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Currents in metric spaces, isoperimetric inequalities, and applications to area minimization problems
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.
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Half-eigenvalues and semilinear problems with jumping nonlinearities
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.
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Complexification phenomenon in a class of singular perturbations
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Cavitation and Configurational Forces in a Nonlinearly Elastic Material
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Ideal Knots
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.
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Elliptic systems, integral functionals and singular sets
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.
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Adaptive finite elements for relaxed methods (FERM) in computational microstructures
Abstract
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Geometric rigidity of conformal matrices
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.
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Polyconvexity and counterexamples to regularity in the calculus of variations
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.
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The Aviles Giga functional
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.
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Symmetry breaking bifurcations, normalized cuts and the neural coding problems
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A variational approach to optimal design
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