Concentration month on nonlinear elliptic PDEs

Organized by Henri Berestycki and Luis Silvestre at the University of Chicago.
These activities are financed by NSF grant DMS-1065979 FRG: Collaborative Research : Emerging Issues in the Sciences Involving Non-Standard Diffusion.
The participants of this grant are Henri Berestycki, Luis Caffarelli, Yanyan Li, Fanghua Lin and Luis Silvestre.

During the month of May 2012, we will have a concentration period involving issues on nonlinear elliptic and parabolic equations with non-standard diffusion and applications.

More details here
In particular, here is the programme

Workshop on Nonlocal operators in Bielefeld (July 2012)

Nonlocal Operators: Analysis, Probability, Geometry and Applications
July 9th – 14th, 2012
ZiF Bielefeld

More details here

In recent years the interest in integro-differential operators and corresponding Markov jump processes has increased significantly. Even for the most simple case, the fractional Laplacian resp. the rotational invariant alpha-stable process, new fundamental results have been proved recently. Areas of research include probability (random walks, Lévy processes), stochastic analysis (jump processes, stochastic differential equations), analysis (potential analysis, linear and fully nonlinear differential operators, variational calculus) and geometry. Moreover, corresponding models are considered in various fields of natural sciences. The aim of this conference is to bring together specialists from different fields with complementary views and to stimulate interaction.

A summer school will preceed this workshop, with Lectures by L. Silvestre and Z.-Q. Chen
see here

Hölder continuity for mixed integro-differential equations (slides)

Here are the slides of the talk I gave during the workshop Nonlocal PDEs, Variational Problems and their Applications.

Abstract: In a joint work with G. Barles, E. Chasseigne and A. Ciomaga, we establish Holder and Lipschitz estimates for solutions of elliptic nonlinear integro-differential equations, by the classical Ishii-Lions’s method. The main novelty lies in the type of nonlocal equations we can deal with: they can be degenerate both in the local and nonlocal term, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one. Such equations are referred to as mixed integro-differential equations.

Nonlocal PDEs in UCLA

Nonlocal PDEs, Variational Problems and their Applications

February 27 – March 2, 2012

Here is the webpage of the conference.
Here is the schedule.

Hamilton-Jacobi equations posed on junctions (slides)

Here are the slides of a talk (see also here) I gave in ACSIOM seminar at Montpellier and in the Numerical analysis seminar at Rennes.

In this talk, I present the results obtained in collaboration with R. Monneau and H. Zidani about Hamilton-Jacobi equation posed on a singular 1D domain, namely a junction. The main result is a comparison principle. The motivation comes from traffic flows.

Abstract (in French): Dans un travail récent en collaboration avec R. Monneau et H. Zidani, nous avons étudié une équation de Hamilton-Jacobi (HJ) posée sur un ensemble de demi-droites dont l’extrémité est commune, par exemple un Y. On appelle cela une jonction. Même dans le cadre le plus simple, si l’on conserve la difficulté d’une éventuelle discontinuité de l’Hamiltonien au point de jonction, il est difficile d’obtenir l’unicité de la solution de l’équation de HJ. Cette difficulté apparaît notamment si ces résultats sont appliqués à des problèmes de trafic routier, motivation première de ce travail.

New preprint about C^{1,α} regularity for degenerate elliptic equations

A new preprint has been uploaded in HAL and arXiv servers! It is a joint work with L. Silvestre about the C^{1,alpha} regularity of solutions of fully nonlinear elliptic equations that are degenerate when the gradient is small. Here is the abstract:

In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. H\”older estimates obtained by the first author (2011) are combined with new Lipschitz estimates obtained through the Ishii-Lions method in order to get C^{1,α} estimates for solutions of these equations.

C’est parti

Bienvenue sur ce blog sur lequel je posterai essentiellement mes nouveaux preprints. On pourra aussi y trouver la liste complète de mes travaux, la liste de mes prépublications présentes sur HAL, de celles sur arXiv, un lien vers la liste de mes publications sur MathSciNet etc. Et puis, sait-on jamais, je prendrai peut-être goût à cette nouvelle forme de communication scientifique.

Suivre

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