Gibson 1st Floor SR

Some results of R.Harvey and B.Lawson on the Dirichlet problem for a class of fully nonlinear elliptic equations will be presented.

No background is required; the talk will be expository.

This work in collaboration with J. Casado-Díaz deals with the asymptotic behaviour of two-dimensional linear conduction problems for which the sequence of conductivity matrices is bounded from below but not necessarily from above. On the one hand, we prove an extension in dimension two of the classical div-curl lemma, which allows us to derive a H-convergence type result for any L¹-bounded sequence of conductivity matrices. On the other hand, we obtain a uniform convergence result satisfied by the minimisers of a sequence of two-dimensional diffusion energies. This implies the closure for the L²-strong topology of $\Gamma$-convergence of the sets of equicoercive diffusion energies without assuming any bound from above. A few counter-examples in dimension three, connected with the appearance of non-local effects, show the specificity of dimension two in the two previous compactness results.

We consider problems of elastic plastic deformation with isotropic and  kinematic hardening.

A dual formulation with stresses as principal variables is used. 

We obtain several results on Sobolev space regularity of the stresses  and strains.

In particular, we obtain the existence of a full derivative of the  stress tensor up to the boundary of the basic domain.

Finally, we present an outlook for obtaining further regularity  results in connection with general nonlinear evolution problems.

We are interested in optimizing the compliance of an elastic structure when the applied forces are partially unknown or submitted to perturbations, the so-called "robust compliance".

For linear elasticity,the compliance is a solution to a minimizing problem of the energy. The robust compliance is then a min-max, the minimum beeing taken amongst the possible displacements and the maximum amongst the perturbations. We show that this problem is well-posed and easy to compute.

We then show that the problem is relatively easy to differentiate with respect to the domain and to compute the steepest direction of descent.

The levelset algorithm is then applied and many examples will explain the different mathematical and technical difficulties one faces when one

tries to tackle this problem.

The aim of my talk is to present the work of my PhD Thesis and my current research. It is concerned with local/global bifurcation of standing wave solutions to some nonlinear Schr\"odinger equations in $\mathbb{R}^N \ (N\geq1)$ and with stability properties of these solutions. The equations considered have a nonlinearity of the form $V(x)|\psi|^{p-1}\psi$, where $V:\mathbb{R}^N\to\mathbb{R}$ decays at infinity and is subject to various assumptions. In particular, $V$ could be singular at the origin.

Local/global smooth branches of solutions are obtained for the stationary equation by combining variational techniques and the implicit function theorem. The orbital stability of the corresponding standing waves is studied by means of the abstract theory of Grillakis, Shatah and Strauss.