Mr Pierre LISSY, Universite Dauphine, (France)

Shape optimization is a part of the field of optimization that concentrates on minimizing cost functionals depending on a ``shape'' (here, a set of R^n for n>= 2). It includes a large number of different class of problems, that might go from theorical to very practical questions. Here, we will focus on a very particular class of problems, where we would like to optimize the eigenvalues of the Dirichet-Laplace operator on the class of open bounded sets of R^n, under certain constraints.

Table of contents :

I - Some complements on spectral theory of Dirichlet-Laplace operators

1°) The Dirichlet-Laplace operator

2°) The Rayleigh quotient and its consequences

3°) The max-min principle for the second eigenvalue and its consequences

II - Optimization of the first eigenvalue

1°) Non-existence of a minimizer without constraints

2°) Existence of a minimizer under a volume constraint

3°) Obstructions to uniqueness

4°) Uniqueness for connected domains, up to sets of capacity 0

III - Optimization of the second eigenvalue

1°) Existence of a minimizer under a volume constraint

2°) Non-existence of a minimizer in the class of connected sets

Mr Valentina FRANCESCHI, Université Paris Sud, (France)

L'un des problèmes plus anciens en Mathématiques est le problème isopérimétrique: il s'agit de comprendre que les sphères sont les formes de périmètre minimale parmi celles qui contiennent un volume donné. Cette relation d’optimalité s’exprime au travers d’une inégalité fonctionnelle dite inégalité isopérimétrique, sujet fondamentales dans nombreux problèmes en géométrie, analyse et physique.

Dans ce cours nous étudions:

1. les inégalités isopérimétriques classiques et ses liens avec d’autres inégalités fonctionnelles fondamentales (e.g., inégalités de Sobolev, estimés du noyau de la chaleur).

2. l'optimalité de la boule et la généralisation aux problèmes isopérimétriques à plus volumes donnés (problèmes de clustering minimale).

3. des problèmes ouvertes dans ce cadre, dans le contexte de la géométrie sous-Riemannienne et des espaces avec densité.

MT Dario PRANDI, Laboratoire des signaux et systèmes, CNRS/École Centrale Supelec/Université Paris-Sud,(France)

In this course we will present a brief introduction to spectral geometry, that is, the study of the relation between the geometry of a manifold and the spectrum of the associated Laplace-Beltrami operator. In particular we will present:

1. Introduction to the Laplace-Beltrami operator on Riemannian manifolds, and its spectrum

2. Weyl’s law, that is, how the simple knowledge of the dimension and the volume of the manifold is enough to recover information about the distribution of eigenvalues for large energy

3. Inverse problems, that is, to what extent is possible to construct manifolds with prescribed spectrum.

Mr Wilfrid GANGBO, University of California, Los Angeles (USA)

This course will be devoted to the fundamental and powerful tools of variational analysis, and will shed light on recent developments in optimal transport theory and calculus of variations which have substantial applications in partial differential equations, geometry, economics and physics

Partial differential equations and variational models play a crucial role in mathematical image processing. This course will begin with a brief overview of two famous uses of such techniques: image reconstruction -- the task of reconstructing an image from deformed and/or indirect measurements; and image segmentation -- the task of splitting an image into its key segments.

For image segmentation, we focus on a highly efficient contemporary technique of using "PDEs on graphs". We provide an introduction to analysis on graphs, discuss how an image can be represented as a graph, and how image segmentation can therefore be reinterpreted as a minimisation problem on a graph. We will describe how to approximate solutions to this minimisation problem by solving a version of the Allen--Cahn equation on the graph, and describe in detail the numerical challenges that arise in solving this graph PDE for real images.

Finally, there has been considerable recent work exploring the continuum limits of variational models and PDEs on graphs. We will discuss how the graph setting relates to corresponding continuum objects, providing novel tools for analysis of these graph-based algorithms.

Mr Guy DEGLA, IMSP (Benin)

The aim of this talk is to stress the concepts of properly posed (well-posed) problems and improperly posed (ill-posed) phenomena for Partial Differential Equations subject to initial or nonlocal Boundary Conditions. Ill-posed problems include blow-up phenomena which depends upon the behavior of steady state solutions in the case of evolution equations.

After recalling how to deal in general with ill-posed problems by using analysis tools such as implicit function theorem, bifurcation theory or regularization methods, we shall show the importance of potential well theory, concavity method or comparison methods in the study of blowup behavior of solutions of evolution equations.

Needless to add that research on ill-posed problems has been intensified during the last decades and has risen to many directions including Inverse Problems, Climatology, Piezoelectric Material Science, Geophysics, Oil Reservoir Engineering and Medicine.

TANNENBAUM, Rina

Complex fluids are materials with dynamic internal components whose structural fluctuations and reorganization have a fundamental impact on the rheological properties of the fluid. Examples of such two-phase materials include polymers solutions and melts, liquid crystals, gels, foams, suspensions, emulsions and micellar solutions. From a fundamental perspective, such complex fluids are extremely interesting. They feature dynamic coupling of three disparate length scales: molecular conformations inside each component, mesoscopic interfacial morphology, and macroscopic hydrodynamics. The dynamics of these systems are highly nonlinear in nature, where the increase in stress by an infinitesimal amount or a small displacement of a single particle can result in the difference between an arrested state and fluid-like behavior. Such materials often have great practical utility since the microstructure can be manipulated via processing of the flow to produce useful mechanical, optical, or thermal properties. As a result, the dynamics of the components comprising the complex fluids are an area of active interdisciplinary research. This short course will explore the application of PDEs to common problems associated with the flow properties of complex fluids. We will discuss the viscosity-dominated flow regimes of these fluids (Hagen-Poiseuille), the thermodynamic characteristics of mixtures (Flory-Huggins) and the concept of the chemical potential, and, if time permits, the diffusion-driven nucleation and growth processes and the development of microstructures.

Mr José A. CARILLO, University of Oxford

We analyze under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and and attraction modelled by nonlocal interaction, occurs. It will discuss the regimes that will appear in aggregation diffusion problems with homogeneous kernels. We will discuss the main qualitative properties in terms of stationary states and minimizers of the free energies. In particular, all the porous medium cases are critical while the fast diffusion are not in this regime. In the second part, I will discuss the diffusion dominated case for porous medium cases in which this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary cases with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrisation techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally in the fast diffusion regime we are able to find conditions for the existence of global minimizers in a different range by means of reverse HLS inequalities. Concentration at the origin in part of this range is not ruled out. This talk is based on works in collaboration with S. Hittmeir, B. Volzone, and Y. Yao (to appear in Inventiones Mathematicae), with V. Calvez and F. Hoffmann (Nonlinear Analysis TMA) and with J. Dolbeault, M. Delgadino, F. Hoffmann and R. Frank (to appear in J. Math. Pure Appl.).

Mr Oliver MENOUKEU PAMEN, AIMS (Ghana)

In this lecture we introduce the notion of path by path uniqueness of a solution to a stochastic differential equation. We then discuss the path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet assuming that the drift coefficient is unbounded, verifies a spacial linear growth condition and is componentwise nondeacreasing. We will also discuss the existence of a unique strong solution of multidimensional SDEs driven by Brownian sheet when the drift is non-decreasing and satisfies a spacial linear growth condition. This lecture is based on join work with Antoine Bogso, Moustapha Dieye.

MOYANO, Ivan

Following Arnold's geometric interpretation, the Euler equations of an incompressible fluid moving in a domain D are known to be the optimality equation of the minimizing geodesic problem along the group of orientation and volume preserving diffeomorphisms of D. This problem admits a well-established convex relaxation which generates a set of "relaxed", "multi-stream", version of the Euler equations. However, it is unclear that such relaxed equations are appropriate for the initial value problem and the theory of turbulence, due to their lack of well-posedness for most initial data. As an attempt to get a more relevant set of relaxed Euler equations, we address the multi-stream pressure-less gravitational Euler-Poisson system as an approximate model, for which we show that the initial value problem can be stated as a concave maximization problem from which we can at least recover a large class of smooth solutions for short enough times. Joint work with Yann Brenier (Orsay).