Lawrence C. Evans

Luis A. Caffarelli

Luigi Ambrosio

Michael G. Crandall

Nicola Fusco

Centro internazionale matematico estivo. Summer School

E. Mascolo

With a historical overview by Elvira Mascolo

Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, June 27 - July 2, 2005

Calculus of Variations and Nonlinear Partial Differential Equations

Elia Brué

Daniele Semola

This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations.

Lectures on Optimal Transport

Nicola Gigli

Giuseppe Savare

This book is devoted to a theory of gradient ?ows in spaces which are not nec- sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 1 devoted to gradient ?ows in the L -Wasserstein space of probability measures on p a separable Hilbert space X (we consider the L -Wasserstein distance, p? (1,?), as well). The two parts have some connections, due to the fact that the Wasserstein space of probability measures provides an important model to which the “metric” theory applies, but the book is conceived in such a way that the two parts can be read independently, the ?rst one by the reader more interested to Non-Smooth Analysis and Analysis in Metric Spaces, and the second one by the reader more oriented to theapplications in Partial Di?erential Equations, Measure Theory and Probability.

In Metric Spaces and in the Space of Probability Measures

Gradient Flows

Alessandro Carlotto

Annalisa Massaccesi

The book originates from the Elliptic PDE course given by the first author at the Scuola Normale Superiore in recent years. It covers the most classical aspects of the theory of Elliptic Partial Differential Equations and Calculus of Variations, including also more recent developments on partial regularity for systems and the theory of viscosity solutions.

Lectures on Elliptic Partial Differential Equations

Andrea Mondino

Giuseppe Savaré

The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.

Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces

Biocomposites are widely used in the medical industry to repair and restore bone, tooth, cartilage skin and other tissues. Biomedical composites, provides a thorough review of the current status, recent progress and future trends in composites for biomedical applications. Part one discusses the fundamentals of biocomposites with chapters on natural composites, design and fabrication of biocomposites, and hard and soft tissue applications of biocomposites. Part two then reviews applications of biocomposites. Chapters discuss composites for bone repair, composite coatings for implants, composites for spinal implants, injectable composites and composites for tissue engineered scaffolds. Chapters in part three discuss the biocompatibility, mechanical behaviour and failure of biocomposites with such topics as cellular response, testing of biocomposites and tribology of biocomposites. Finally part four reviews the future for biocomposites with chapters on nano-structured biocomposites, developing biocomposites as scaffolds and biocomposites in tissue engineering and regenerative medicine. With its distinguished editor and team of international contributors, Biomedical composites is an essential reference to materials scientists and researchers in industry and academia, as well as all those concerned with this increasingly important field. Provides a thorough review of the current status, recent progress and future trends in composites for biomedical applications Discusses the fundamentals of biocomposites with chapters on natural composites, design and fabrication of biocomposites and their applications Chapters address composites for bone repair, spinal implants and various other applications and discuss biocompatability, mechanical behaviour and failure of biocomposites

Biomedical Composites

Dirk Helbing

Alberto Bressan

Axel Klar

Enrique Zuazua

In recent years flows in networks have attracted the interest of many researchers from different areas, e.g. applied mathematicians, engineers, physicists, economists. The main reason for this ubiquity is the wide and diverse range of applications, such as vehicular traffic, supply chains, blood flow, irrigation channels, data networks and others. This book presents an extensive set of notes by world leaders on the main mathematical techniques used to address such problems, together with investigations into specific applications. The main focus is on partial differential equations in networks, but ordinary differential equations and optimal transport are also included. Moreover, the modeling is completed by analysis, numerics, control and optimization of flows in networks. The book will be a valuable resource for every researcher or student interested in the subject.

Cetraro, Italy 2009, Editors: Benedetto Piccoli, Michel Rascle

Modelling and Optimisation of Flows on Networks

The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.

Biomedical Composites, Second Edition, provides revised, expanded, and updated content suitable for those active in the biomaterials and bioengineering field. Three new chapters cover modeling of biocomposites, 3D printing of customized scaffolds, and constructs and regulatory issues. Chapters from the first edition have been revised in order to provide up-to-date, comprehensive coverage of developments in the field. Part One discusses the fundamentals of biocomposites, with Part Two detailing a wide range of applications of biocomposites. Chapters in Part Three discuss the biocompatibility, mechanical behavior, and failure of biocomposites, while the final section looks at the future for biocomposites. Professor Luigi Ambrosio is the Director of the Institute for Composite and Biomedical Materials, Italy. He is a renowned scientist with expertise in biomedical composites and has published over 150 papers in international scientific journals and books, 16 patents, and over 250 presentations at international and national conferences. Led by an expert editor with many years of experience in academia and widely recognized as an international expert on biomedical composites Features an overview of biocomposites for a wide range of biomedical applications Provides revised, expanded, and updated coverage, including three new chapters

Cédric Villani

Giuseppe Buttazzo

Yann Brenier

Leading researchers in the field of Optimal Transportation, with different views and perspectives, contribute to this Summer School volume: Monge-Ampère and Monge-Kantorovich theory, shape optimization and mass transportation are linked, among others, to applications in fluid mechanics granular material physics and statistical mechanics, emphasizing the attractiveness of the subject from both a theoretical and applied point of view. The volume is designed to become a guide to researchers willing to enter into this challenging and useful theory.

Lectures Given at the C.I.M.E. Summer School Held in Martina Franca, Italy, September 2-8, 2001

Optimal Transportation and Applications

Vincenzo Guarino

Electrofluidodynamic Technologies (EFDTs) for Biomaterials and Medical Devices: Principles and Advances focuses on the fundamentals of EFDTs - namely electrospinning, electrospraying and electrodynamic atomization - to develop active platforms made of synthetic or natural polymers for use in tissue engineering, restoration and therapeutic treatments. The first part of this book deals with main technological aspects of EFDTs, such as basic technologies and the role of process parameters. The second part addresses applications of EFDTs in biomedical fields, with chapters on their application in tissue engineering, molecular delivery and implantable devices. This book is a valuable resource for materials scientists, biomedical engineers and clinicians alike. Presents a complete picture of Electrofluidodynamic technologies and their use in biomedicine Provides a comprehensive, professional reference on the subject, covering materials processing, fabrication and the use of novel devices for tissue engineering and therapeutics Focuses on technological advances, with an emphasis on studies and clinical trials

Luigi Ambrosio

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