Nonlinear PDEs and Applications [electronic resource] : C.I.M.E. Summer School, Cetraro, Italy 2008, Editors: Luigi Ambrosio, Giuseppe SavarȨ / by Stefano Bianchini, Eric A. Carlen, Alexander Mielke, CȨdric Villani.

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.

This volume collects the notes of the CIME course "Nonlinear PDEs 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 CȨdric 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 andpoints to new research directions in contributions by leading experts in these fields.

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