This volume collects the notes of the CIME course "Nonlinear PDE's and applications" held in Cetraro (Italy) on June 2328, 2008. It consists of four series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cédric Villani (Ecole Normale Superieure de Lyon). They presented a broad overview of far-reaching findings and exciting new developments concerning, in particular, optimal transport theory, nonlinear evolution equations, functional inequalities, and differential geometry. A sampling of the main topics considered here includes optimal transport, Hamilton-Jacobi equations, Riemannian geometry, and their links with sharp geometric/functional inequalities, variational methods for studying nonlinear evolution equations and their scaling properties, and the metric/energetic theory of gradient flows and of rate-independent evolution problems. The book explores the fundamental connections between all of these topics and points to new research directions in contributions by leading experts in these fields.

A broad overview on some deep results and new exciting developments Contribution of leading experts Accessible introduction to new theories and results

Autorentext
Cédric Villani, geboren 1973, gehört zu den weltweit führenden und innovativsten Mathematikern. Für seine Forschung erhielt er mehrere Preise, u.a. 2010 die renommierte Fields-Medaille, das Äquivalent zum Nobelpreis. Villani ist Professor für Mathematik an der École Normale Supérieure in Lyon und Direktor des Institut Henri Poincaré in Paris.

Inhalt
Transport Rays and Applications to Hamilton-Jacobi Equations.- Functional Inequalities and Dynamics.- Differential, Energetic, and Metric Formulations for Rate-independent Processes.- Optimal Transport and Curvature.