Gibson 1st Floor SR

In this talk we discuss the solution of the elastodynamic

equations in a bounded domain with hereditary-type linear

viscoelastic constitutive relation. Existence, uniqueness, and

regularity of solutions to this problem is demonstrated

for those viscoelastic relaxation tensors satisfying the condition

of being completely monotone. We then consider the non-self-adjoint

and non-linear eigenvalue problem associated with the

frequency-domain form of the elastodynamic equations, and show how

the time-domain solution of the equations can be expressed in

terms of an eigenfunction expansion.

We consider a 3-dimensional elastic continuum whose material points

can experience no displacements, only rotations. This framework is a

special case of the Cosserat theory of elasticity. Rotations of

material points of the continuum are described mathematically by

attaching to each geometric point an orthonormal basis which gives a

field of orthonormal bases called the coframe. As the dynamical

variables (unknowns) of our theory we choose the coframe and a

density.

In the first part of the talk we write down the general dynamic

variational functional of our problem. In doing this we follow the

logic of classical linear elasticity with displacements replaced by

rotations and strain replaced by torsion. The corresponding

Euler-Lagrange equations turn out to be nonlinear, with the source

of this nonlinearity being purely geometric: unlike displacements,

rotations in 3D do not commute.

In the second part of the talk we present a class of explicit

solutions of our Euler-Lagrange equations. We call these solutions

plane waves. We identify two types of plane waves and calculate

their velocities.

In the third part of the talk we consider a particular case of our

theory when only one of the three rotational elastic moduli, that

corresponding to axial torsion, is nonzero. We examine this case in

detail and seek solutions which oscillate harmonically in time but

depend on the space coordinates in an arbitrary manner (this is a

far more general setting than with plane waves). We show [1] that

our second order nonlinear Euler-Lagrange equations are equivalent

to a pair of linear first order massless Dirac equations. The

crucial element of the proof is the observation that our Lagrangian

admits a factorisation.

[1] Olga Chervova and Dmitri Vassiliev, "The stationary Weyl

equation and Cosserat elasticity", preprint http://arxiv.org/abs/1001.4726

Numerical homogenization/upscaling for problems with multiple scales have attracted increasing attention in recent years. In particular, problems with non-separable scales pose a great challenge to mathematical analysis and simulation.

In this talk, we present some rigorous results on homogenization of divergence form scalar and vectorial elliptic equations with $L^\infty$ rough coefficients which allow for a continuum of scales. The first approach is based on a new type of compensation phenomena for scalar elliptic equations using the so-called ``harmonic coordinates''. The second approach, the so-called ``flux norm approach'' can be applied to finite dimensional homogenization approximations of both scalar and vectorial problems with non-separated scales. It can be shown that in the flux norm, the error associated with approximating the set of solutions of the PDEs with rough coefficients, in a properly defined finite-dimensional basis, is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard finite element space. We will also talk about the ongoing work on the localization of the basis in the flux norm approach.

After hep-th/0909.0483

The aim of this talk is to analyze energy functionals concentrated on the jump set of 2D vector fields of unit length and of vanishing divergence.

The motivation of this study comes from thin-film micromagnetics where these functionals correspond to limiting wall-energies. The main issue consists in characterizing the wall-energy density (the cost function) so that the energy functional is lower semicontinuous (l.s.c.). The key point resides in the concept of entropies due to the scalar conservation law implied by our vector fields. Our main result identifies appropriate cost functions

associated to certain sets of entropies. In particular, certain power cost functions lead to l.s.c. energy functionals.

A second issue concerns the existence of minimizers of such energy functionals that we prove via a compactness result. A natural question is whether the viscosity solution is a minimizing configuration. We show that in general it is not the case for nonconvex domains.

However, the case of convex domains is still open. It is a joint work with Benoit Merlet, Ecole Polytechnique (Paris).

Based on hep-th/0912.3249 by Arkani-Hamed et. al..

Let $\phi$ be a convex, $C^1$-function and consider the functional: $$ (1)\qquad \mathcal{F}(\bf u)=\int_{\Omega} \phi (|\nabla \bf u|) \,dx $$ where $\Omega\subset \mathbb{R}^n$ is a bounded open set and $\bf u: \Omega \to \mathbb{R}^N$. The associated Euler Lagrange system is $$ -\mbox{div} (\phi' (|\nabla\bf u|)\frac{\nabla\bf u}{|\nabla\bf u|} )=0 $$ In a fundamental paper K.~Uhlenbeck proved everywhere $C^{1,\alpha}$-regularity for local minimizers of the $p$-growth functional with $p\ge 2$. Later on a large number of generalizations have been made. The case $1

{\bf Theorem.} Let $\bfu\in W^{1,\phi}_{\loc}(\Omega)$ be a local minimizer of (1), where $\phi$ satisfies suitable assumptions. Then $\bfV(\nabla \bfu)$ and $\nabla \bfu$ are locally $\alpha$-Hölder continuous for some $\alpha>0$.

We present a unified approach to the superquadratic and subquadratic $p$-growth, also considering more general functions than the powers. As an application, we prove Lipschitz regularity for local minimizers of asymptotically convex functionals in a $C^2$ sense.

This is a review of hep-th/0912.4912