In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change.

Résumé

Dans cet exposé, on rappelle le problème de la tomographie et sa résolution par des méthodes de type inversion de la transformée de Radon. En particulier, la célèbre méthode de Feldkamp, Davis et Kress (FDK) est présentée. Puis, nous interprétons l’usage de cette méthode en imagerie laser 3D, avec résultats numériques sur des données industrielles à  l’appui.

In this talk, starting from the paper of R. Brizzi et J.P. Chalot on problems in domains with strongly oscillating boundaries, I shall recall the contribution with D. Blanchard on this subject and I shall present recent results obtained with O. Guibé.

A domain with strongly oscillating boundary is a domain whose boundary
presents numerous asperities. The asperities have fixed height, a size
depending on a small parameter $varepsilon$ and $varepsilon$-periodic
structure.

Boundary-value problems in such a domain arise in many fields of biology,
physics and engineering sciences. It is often impossible to approach these
problems directly with numerical methods, because the rough boundary
requires a large number of mesh points in its neighborhood. Thus, the
computational cost associated to such a problem grows rapidly when
$varepsilon$ gets smaller. Moreover, it can occur that the required
discretization step becomes too small for the machine precision. Then, the
goal is to replace the problem, when the periodicity $varepsilon$ gets
smaller, with a model in a “more regular” domain $Omega$ which can be
numerically solved.

On étudie l’existence de solutions de certains problèmes faisant intervenir l’exposant critique de Sobolev et avec un poids. On étudie également la multiplicité des solutions du problème dans le cas o๠l’ensemble des minima du poids admet plusieurs composantes connexes. On s’intéresse aussi au cas o๠cet ensemble possède une seule composante connexe et une topologie complexe.

Dans cet exposé je présenterai différents résultats concernant l’existence (ou la non-existence) de valeurs propres discrètes pour le Laplacien défini sur un domaine tubulaire donné à  partir d’un guide droit déformé par torsion le long son axe longitudinal.

We study particulate flows or suspensions of solid particles in a viscous incompressible fluid in the presence of highly oscillatory, velocity dependent, surface forces. The flow at a small Reynolds number is modeled by the Stokes equations coupled with the motion of rigid particles. The objective is to perform homogenization for the given suspension and obtain an equivalent description of a homogeneous (effective) medium and determine the effective viscosity and the macroscopic effect of the surface forces.