Recent progress in Nonlinear Analysis

Journée thématique

"Recent progress in Nonlinear Analysis"

Marseille, 10-11 décembre 2012

Conférenciers invités

    Prof. Bernhard Ruf (Milano)
    Prof. Tobias Weth (Frankfurt)

Programme (mise à jour 10/12/12 18h00)

Le workshop, qui consiste de deux mini-cours de la durée totale de 2h30 chacun, se déroulera dans la salle de conférences de la FRUMAM (Site St.Charles - Batiment Chimie - 3e étage - plan d'accès).


"Borderline cases of Sobolev embeddings and related nonlinear PDE's" (Bernhard Ruf) (pdf version)

The well-known Sobolev inequalities are a powerful tool in the study of nonlinear PDE's. Of particular interest are the limiting, or so-called critical, embeddings which present several interesting phenomena like: loss of compactness and of solvability, appearance of group invariances, concentration phenomena and a related quantization effect (the so-called bubbling of spheres). The classical Sobolev inequalities have been extended and generalized in many directions. An important class concerns the so-called borderline cases in which exponential growth functions appear. While the classical Sobolev embeddings are by now well understood, there are still many open questions in the borderline cases; in fact, these are more delicate to handle due to a lack of explicit scale invariances, no analogue of the so-called Pohozaev identity, and less friendly spaces to work with. In these lectures, we will describe recent results about the where $\Omega$ denotes a domain in $\mathbb R^n$. We will also present some new results concerning associated nonlinear elliptic equations with critical and supercritical growth.

"Liouville type theorems for a class of non-cooperative elliptic systems" (Tobias Weth)

I will report on recent joint work with Norman Dancer, Hugo Tavares, Susanna Terracini and Gianmaria Verzini. For a class of semilinear elliptic systems with homogeneous gradient type nonlinearity, we study the question of existence and nonexistence of nontrivial nonnegative solutions on the entire space and for the corresponding half space Dirichlet problem. Our results imply - for certain parameter values - a priori bounds for more general boundary value problems which we also discuss. Our study includes weakly coupled Schrödinger systems which have attracted great interest in recent years due to their appearance in nonlinear optics and in models for multi-component mixtures of Bose-Einstein condensates. In some cases, we identify optimal parameter ranges for which a priori bounds hold. The key difference to classical results is that the systems we consider are non-cooperative.

Last modified: 25/11/2012