Geometric Analysis and Nonlinear Partial Differential Equations

ISBN-13: 9783642628870 / Angielski / Miękka / 2013 / 673 str.

Stefan Hildebrandt; Hermann Karcher

Opis

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."

Spis treści

Olga Ladyzhenskaya—A Life-Long Devotion to Mathematics.- I. Geometric Analysis and Calculus of Variations.- On the Spectral Theory of Surfaces with Cusps.- The Dirac Determinant of Spherical Space Forms.- Constructing Isospectral Metrics via Principal Connections.- Parametrizations of Teichmüller Space and Its Thurston Boundary.- Linearization of Isotropic Automorphisms of Non-quadratic Elliptic CR-Manifolds in ?4.- Global C2+?-Estimates for Conformai Maps.- On Karcher’s Twisted Saddle Towers.- Unstable Periodic Discrete Minimal Surfaces.- An Adaptive Finite Element Method for Minimal Surfaces.- Singular Minimal Surfaces.- Note on the Isoperimetric Profile of a Convex Body.- Geometric Conditions on Free Boundaries.- On Generalized Mean Curvature Flow in Surface Processing.- A Finite Element Level Set Method for Anisotropic Mean Curvature Flow with Space Dependent Weight.- Optimal Regularity Results via A-Harmonic Approximation.- Dominance Functions for Parametric Lagrangians.- Convex Variational Problems with Linear Growth.- II. Nonlinear Partial Differential Equations.- Studying Nonlinear pde by Geometry in Matrix Space.- On the Korteweg — de Vries Equation and KAM Theory.- Convergence of Approximate Solutions of Conservation Laws.- Nonlinear Hyperbolic Systems of Generalized Navier-Stokes Type for Interactive Motion in Biology.- On Peak and Periodic Solutions of an Integro-Differential Equation on S1.- Symmetrizing Measures for Infinite Dimensional Diffusions: An Analytic Approach.- Markov Semigroups and Harmonic Maps.- Boundary Regularity for Nonlinear Elliptic Systems: Applications to the Transmission Problem.- A Particle-Partition of Unity Method — Part V: Boundary Conditions.- On Uniqueness- and Regularity Criteria for the Navier-Stokes Equations.- Problems Due to the No-Slip Boundary in Incompressible Fluid Dynamics.- Comparison of Finite Volume and Discontinuous Galerkin Methods of Higher Order for Systems of Conservation Laws in Multiple Space Dimensions.- Existence of Strong Solutions for Electrorheological Fluids in Two Dimensions: Steady Dirichlet Problem.- Spinodal Decomposition in the Presence of Elastic Interactions.- Waiting Time Phenomena for Degenerate Parabolic Equations — A Unifying Approach.- The Mathematics of Ostwald Ripening.- Appendix. Color Plates.