The week will focus on various aspects of functional inequalities, and their link with entropy methods in PDE and probability.

Two minicourses are scheduled : Jean Dolbeault and Eric Carlen.

Talks by invited speakers and working sessions are scheduled. (S. Bobkov, E. Millman, F. Bolley, X. Li)

**Eric Carlen** :

**Title**: Functional inequalities and evolution equations

**Abstract** :

Functional inequalities are an essential tool for understanding the behavior of solutionsof PDE's and other sorts of evolution equations. At the same time, monotonicity along various evolution processes can often be used to prove functional inequalities. This course will focus on some recent examples in which this interplay between evolution processes and functional inequalities has been fruitful, and it will also introduce some open problems in the field.

**Jean Dolbeault** :

**Title**: Sharp functional inequalities and nonlinear flows

**Abstract** :

The lectures will be devoted to functionals inequalities related to Sobolev's inequality, the associated sharp constants, and the strategies developed to obtain so-called improved inequalities. The main tools employed are linear or nonlinear flows associated to an evolutionary partial differential equation (often a diffusion equation or a drift-diffusion equation). More precisely, one has to find out a functional (generalised entropy, free energy) related to the sought-for inequality, which decreases during time evolution, that is, a Lyapunov functional. On the one hand, the main issues investigated will be related to the structure of the problems (when is the evolution equation the gradient flow of the entropy for a suitably chosen distance? When is the flow endowed with a duality structure?) and to optimality questions (when does an improved version of some inequality guarantee that equality is realized in that inequality in the asymptotic regime?). On the other hand, various geometrical situations will be considered (such as manifolds with changing-sign curvature) and a connection to rigidity techniques for semilinear elliptic equations and spectral estimates for Schrödinger operators will be made. In all the above-mentioned problems, the monotonicity property enjoyed by the flows under study does not follow from a pointwise positivity condition but rather from an integral condition, in particular a positivity condition on an eigenvalue.

### Speakers

- Serguei Bobkov (University of Minnesota)
- François Bolley (CEREMADE, Université Paris-Dauphine)
- Eric Carlen (Department of Mathematics, Rutgers University)
- Jean Dolbeault (CEREMADE, Université Paris-Dauphine)
- Emmanuel Milman (Technion University, Haifa)
- Xiangdong Li (Chinese Academy of Sciences)