This book is not a textbook, but rather a coherent collection of papers from the field of partial differential equations. Nevertheless we believe that it may very well serve as a good introduction into some topics of this classical field of analysis which, despite of its long history, is highly modem and well prospering. Richard Courant wrote in 1950: "It has always been a temptationfor mathematicians to present the crystallized product of their thought as a deductive general theory and to relegate the individual mathematical phenomenon into the role of an example. The reader who submits to the dogmatic form will be easily indoctrinated. Enlightenment, however, must come from an understanding of motives; live mathematical development springs from specific natural problems which can be easily understood, but whose solutions are difficult and demand new methods or more general significance. " We think that many, if not all, papers of this book are written in this spirit and will give the reader access to an important branch of analysis by exhibiting interest ing problems worth to be studied. Most of the collected articles have an extensive introductory part describing the history of the presented problems as well as the state of the art and offer a well chosen guide to the literature. This way the papers became lengthier than customary these days, but the level of presentation is such that an advanced graduate student should find the various articles both readable and stimulating.

This volume contains lecture notes on key topics in geometric analysis, a growing mathematical subject which uses analytical techniques, mostly of partial differential equations, to treat problems in differential geometry and mathematical physics.

This book emphasizes the interdisciplinary interaction in problems involving geometry and partial differential equations. It provides an attempt to follow certain threads that interconnect various approaches in the geometric applications and influence of partial differential equations. A few such approaches include: Morse-Palais-Smale theory in global variational calculus, general methods to obtain conservation laws for PDEs, structural investigation for the understanding of the meaning of quantum geometry in PDEs, extensions to super PDEs (formulated in the category of supermanifolds) of the geometrical methods just introduced for PDEs and the harmonic theory which proved to be very important especially after the appearance of the Atiyah-Singer index theorem, which provides a link between geometry and topology.

Choice Outstanding Title! (January 2006) This richly illustrated text covers the Cauchy and Neumann problems for the classical linear equations of mathematical physics. A large number of problems are sprinkled throughout the book, and a full set of problems from examinations given in Moscow are included at the end. Some of these problems are quite challenging! What makes the book unique is Arnold's particular talent at holding a topic up for examination from a new and fresh perspective. He likes to blow away the fog of generality that obscures so much mathematical writing and reveal the essentially simple intuitive ideas underlying the subject. No other mathematical writer does this quite so well as Arnold.

This book describes very recent results involving an extensive use of analytical tools in the study of geometrical and topological properties of complete Riemannian manifolds. It analyzes in detail an extension of the Bochner technique to the non compact setting, yielding conditions which ensure that solutions of geometrically significant differential equations either are trivial (vanishing results) or give rise to finite dimensional vector spaces (finiteness results). The book develops a range of methods, from spectral theory and qualitative properties of solutions of PDEs, to comparison theorems in Riemannian geometry and potential theory.

This book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.

This edited volume has a two-fold purpose. First, comprehensive survey articles provide a way for beginners to ease into the corresponding sub-fields. These are then supplemented by original works that give the more advanced readers a glimpse of the current research in geometric analysis and related PDEs. The book is of significant interest for researchers, including advanced Ph.D. students, working in geometric analysis. Readers who have a secondary interest in geometric analysis will benefit from the survey articles. The results included in this book will stimulate further advances in the subjects: geometric analysis, including complex differential geometry, symplectic geometry, PDEs with a geometric origin, and geometry related to topology. Contributions by Claudio Arezzo, Alberto Della Vedova, Werner Ballmann, Henrik Matthiesen, Panagiotis Polymerakis, Sun-Yung A. Chang, Zheng-Chao Han, Paul Yang, Tobias Holck Colding, William P. Minicozzi II, Panagiotis Dimakis, Richard Melrose, Akito Futaki, Hajime Ono, Jiyuan Han, Jeff A. Viaclovsky, Bruce Kleiner, John Lott, Sławomir Kołodziej, Ngoc Cuong Nguyen, Chi Li, Yuchen Liu, Chenyang Xu, YanYan Li, Luc Nguyen, Bo Wang, Shiguang Ma, Jie Qing, Xiaonan Ma, Sean Timothy Paul, Kyriakos Sergiou, Tristan Rivière, Yanir A. Rubinstein, Natasa Sesum, Jian Song, Jeffrey Streets, Neil S. Trudinger, Yu Yuan, Weiping Zhang, Xiaohua Zhu and Aleksey Zinger.