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This volume collects the notes of the CIME course "Nonlinear PDE s
and applications" held in Cetraro (Italy) on June 23 28, 2008. It
consists of four series of lectures, delivered by Stefano Bianchini
(SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander
Mielke (WIAS, Berlin), and Cedric 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.
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