Ebook: 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.

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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.

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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.

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