Geometric Analysis and Nonlinear Partial Differential Equations

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. Front Matter....Pages I-IX Olga Ladyzhenskaya....Pages 1-10 Front Matter....Pages 11-11 On the Spectral Theory of Surfaces with Cusps....Pages 13-37 The Dirac Determinant of Spherical Space Forms....Pages 39-67 Constructing Isospectral Metrics via Principal Connections....Pages 69-79 Parametrizations of Teichmüller Space and Its Thurston Boundary....Pages 81-88 Linearization of Isotropic Automorphisms of Non-quadratic Elliptic CR-Manifolds in C 4 ....Pages 89-103 Global C 2+α -Estimates for Conformai Maps....Pages 105-115 On Karcher’s Twisted Saddle Towers....Pages 117-127 Unstable Periodic Discrete Minimal Surfaces....Pages 129-145 An Adaptive Finite Element Method for Minimal Surfaces....Pages 147-175 Singular Minimal Surfaces....Pages 177-193 Note on the Isoperimetric Profile of a Convex Body....Pages 195-200 Geometric Conditions on Free Boundaries....Pages 201-216 On Generalized Mean Curvature Flow in Surface Processing....Pages 217-248 A Finite Element Level Set Method for Anisotropic Mean Curvature Flow with Space Dependent Weight....Pages 249-264 Optimal Regularity Results via A -Harmonic Approximation....Pages 265-296 Dominance Functions for Parametric Lagrangians....Pages 297-326 Convex Variational Problems with Linear Growth....Pages 327-344 Front Matter....Pages 345-345 Studying Nonlinear pde by Geometry in Matrix Space....Pages 347-395 On the Korteweg — de Vries Equation and KAM Theory....Pages 397-416 Front Matter....Pages 345-345 Convergence of Approximate Solutions of Conservation Laws....Pages 417-430 Nonlinear Hyperbolic Systems of Generalized Navier-Stokes Type for Interactive Motion in Biology....Pages 431-461 On Peak and Periodic Solutions of an Integro-Differential Equation on S 1 ....Pages 463-474 Symmetrizing Measures for Infinite Dimensional Diffusions: An Analytic Approach....Pages 475-486 Markov Semigroups and Harmonic Maps....Pages 487-504 Boundary Regularity for Nonlinear Elliptic Systems: Applications to the Transmission Problem....Pages 505-517 A Particle-Partition of Unity Method Part V: Boundary Conditions....Pages 519-542 On Uniqueness- and Regularity Criteria for the Navier-Stokes Equations....Pages 543-557 Problems Due to the No-Slip Boundary in Incompressible Fluid Dynamics....Pages 559-571 Comparison of Finite Volume and Discontinuous Galerkin Methods of Higher Order for Systems of Conservation Laws in Multiple Space Dimensions....Pages 573-589 Existence of Strong Solutions for Electrorheological Fluids in Two Dimensions: Steady Dirichlet Problem....Pages 591-602 Spinodal Decomposition in the Presence of Elastic Interactions....Pages 603-635 Waiting Time Phenomena for Degenerate Parabolic Equations — A Unifying Approach....Pages 637-648 The Mathematics of Ostwald Ripening....Pages 649-663 Back Matter....Pages 665-673

this Well-organized And Coherent Collection Of Papers Leads The Reader To The Frontiers Of Present Research In The Theory Of Nonlinear Partial Differential Equations And The Calculus Of Variations And Offers Insight Into Some Exciting Developments. In Addition, Most Articles Also Provide An Excellent Introduction To Their Background, Describing Extensively As They Do The History Of Those Problems Presented, As Well As The State Of The Art And Offer A Well-chosen Guide To The Literature.

part I Contains The Contributions Of Geometric Nature: From Spectral Theory On Regular And Singular Spaces To Regularity Theory Of Solutions Of Variational Problems.

part Ii Consists Of Articles On Partial Differential Equations Which Originate From Problems In Physics, Biology And Shastics. They Cover Elliptic, Hyperbolic And Parabolic Cases.

This well-organized and coherent collection of papers leads the reader to the frontiers of present research in the theory of nonlinear partial differential equations and the calculus of variations and offers insight into some exciting developments. In addition, most articles also provide an excellent introduction to their background, describing extensively as they do the history of those problems presented, as well as the state of the art and offer a well-chosen guide to the literature. Part I contains the contributions of geometric nature: From spectral theory on regular and singular spaces to regularity theory of solutions of variational problems. Part II consists of articles on partial differential equations which originate from problems in physics, biology and stochastics. They cover elliptic, hyperbolic and parabolic cases.