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).
- lundi 10 décembre

- 13h45-15h00: cours de Bernhard Ruf

- 15h30-16h45: cours de Bernhard Ruf

- mardi 11 décembre

- 9h15-10h30: cours de Tobias Weth

- 11h00-12h15: cours de Tobias Weth

## Abstracts

**"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
- Trudinger-Moser inequality, concerning the Sobolev spaces $W^{1,n}(\Omega)$

- Adams inequality, concerning the spaces: $W^{k,\frac n k}(\Omega)$

- Brezis-Merle inequality, concerning the space $W^{2,1}(\Omega)$

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