Des équations aux dérivées partielles au calcul scientifique

Titres et résumés des conférences

 Giovanni Alberti (Università di Pisa) ( paper) Structure of null sets in Euclidean spaces: results and open problems I will describe a decomposition theorem for sets of measure zero in the plane which can be deduced from an elementary combinatorial result (Dillworth's lemma). I will then outline a few applications to elementary questions in Geometric Measure Theory. The extension of these results to higher dimensions is mostly open. This is joint work with D. Preiss (Warwick) and M. Csörnyei (University College London). Abbas Bahri (Rutgers University) Contact homology via Legendrian curves: definition and first steps of computation Using the Legendrian curves associated to a "dual" form, we have defined in an earlier work a contact homology for three dimensional contact structures. We will establish in this talk that this definition holds under full generality. We will show that compactness holds in this framework and we will provide the value of this homology for odd generators. Yann Brenier (Université de Nice) On the field-matter interaction in electromagnetism: a weak convergence approach It has been a long standing goal of theoretical physics to derive interaction between field and matter directly from field equations, without any coupling, matter being interpreted as singularities of the field (such as vortices, for instance) or, alternately, as zones of high intensity of the field (when the field equations do not generate singularities). In the case of classical electromagnetism, this idea has been introduced by Gustav Mie, then followed by Max Born and many others. Starting from the nonlinear theory of Born and Infeld, we will see how the field-matter interaction follows from the original field equations (the Born-Infeld system) after performing two steps: i) adding the conservation of energy-momentum equations to the original field equations, and ii) introducing the weak closure of the algebraic manifold in which the solutions take their value. We will also discuss, with a quite different point of view, a concept of viscosity solutions in the large for this kind of nonlinear systems. Luis Caffarelli (University of Texas at Austin) Fully nonlinear equations for integral diffusions We will present the elements of a regularity theory (ABP theorem, estimates, Harnack type inequality) for "bounded measurable" and "fully nonlinear" diffusions of fractional type. This is a joint project with Luis Silvestre. Constantine Dafermos (Brown University) Hyperbolic balance laws with dissipation The lecture will characterize source terms of hyperbolic systems of balance laws that induce dissipation, in the context of BV solutions, and will discuss implications to the theory of relaxation. Maria Esteban (Université Paris Dauphine) Hardy-Dirac inequalities and applications In this talk I will present various versions of Hardy-like inequalities for the Dirac operator. Particular attention will be given to the case of multipolar and magnetic potentials, where the physical model behind the inequalities as well as their implications will be described in details. Craig Evans (University of California at Berkeley) A nonlinear partial differential equation model for lakes and rivers I will discuss a simple fast diffusion partial differential equation that predicts the formation of "lakes" and "rivers" on a given landscape. Nicola Fusco (Università di Napoli) ( paper) The sharp Sobolev inequality in quantitative form We present a quantitative version of the sharp Sobolev inequality in , 1