The Geometry of Domains in Space

"The book can be highly recommended for graduate students as a comprehensive introduction to the field of geometric analysis. Also mathematicians working in other areas can profit a lot from this carefully written book. In particular, the geometric ideas are presented in a self-contained manner; for some of the needed analytic or measure-theoretic results, references are given."

-ZAA

The analysis of Euclidean space is well-developed. The classical Lie groups that act naturally on Euclidean space-the rotations, dilations, and trans lations-have both shaped and guided this development. In particular, the Fourier transform and the theory of translation invariant operators (convolution transforms) have played a central role in this analysis. Much modern work in analysis takes place on a domain in space. In this context the tools, perforce, must be different. No longer can we expect there to be symmetries. Correspondingly, there is no longer any natural way to apply the Fourier transform. Pseudodifferential operators and Fourier integral operators can playa role in solving some of the problems, but other problems require new, more geometric, ideas. At a more basic level, the analysis of a smoothly bounded domain in space requires a great deal of preliminary spadework. Tubular neighbor hoods, the second fundamental form, the notion of "positive reach", and the implicit function theorem are just some of the tools that need to be invoked regularly to set up this analysis. The normal and tangent bundles become part of the language of classical analysis when that analysis is done on a domain. Many of the ideas in partial differential equations-such as Egorov's canonical transformation theorem-become rather natural when viewed in geometric language. Many of the questions that are natural to an analyst-such as extension theorems for various classes of functions-are most naturally formulated using ideas from geometry.

Autorentext
Steven Krantz, Ph.D., is Chairman of the Mathematics Department at Washington University in St. Louis. An award-winning teacher and author, Dr. Krantz has written more than 45 books on mathematics, including Calculus Demystified, another popular title in this series. He lives in St. Louis, Missouri.

Klappentext
Wherein classical analysis is based on the study of the classical groups, the modern setting, which is the focus of this work, treats the subject of domains in space. These domains are the natural venue for various problems that arise from physics, and are also the natural setting for the boundary value problems of PDEs.

Zusammenfassung
The analysis of Euclidean space is well-developed. In particular, the Fourier transform and the theory of translation invariant operators (convolution transforms) have played a central role in this analysis. The normal and tangent bundles become part of the language of classical analysis when that analysis is done on a domain.

Inhalt

1. Elementary Topics.- 1.1 Smooth Functions.- 1.2 The Concept of Defining Function.- 1.3 Measure Theory.- 2. Domains with Smooth Boundaries.- 2.1 The Tangent Bundle and Normal Bundle of the Boundary.- 2.2 The Second Fundamental Form and Curvature.- 2.3 Surfaces with Constant Mean Curvature.- 3. Measures.- 3.1 The Carathéodory Construction.- 3.2 Rectifiability.- 3.3 Minkowski Content.- 3.4 A Space-Filling Curve.- 3.5 Covering Lemmas.- 3.6 Functions of Bounded Variation.- 3.7 Domains with Finite Perimeter.- 3.8 The Area Formula.- 3.9 The Co-Area Formula.- 4. Sobolev Spaces.- 4.1 Basic Definitions and Results.- 4.2 Restriction and Trace Theorems for Sobolev Spaces.- 4.3 Domain Extension Theorems for Sobolev Spaces.- 5. Smooth Mappings.- 5.1 Sard's Theorem.- 5.2 Extension Theorems.- 5.3 Proof of the Whitney Extension Theorem.- 5.4 Application of the Whitney Extension Theorem.- 5.5 Multidimensional Versions of the Fundamental Theorem of Calculus.- 6. Convexity.- 6.1 The Classical Notion of Convexity.- 6.2 Other Characterizations of Convexity.- 6.3 Exhaustion Functions.- 6.4 Convexity of Order k.- 7. Steiner Symmetrization.- 7.1 Basic Properties.- 7.2 The Isodiametric, Isoperimetric and Brunn-Minkowski Inequalities.- 7.3 Equality in the Isoperimetric Inequality.- 8. Topics Related to Complex Analysis.- 8.1 Quasiconformal Mappings.- 8.2 Weyl's Theorem on Eigenvalue Asymptotics of a Domain in Space.- A.1. Metrics on the Collection of Subsets of Euclidean Space.- A.2. The Constants Associated to Euclidean Space.- Guide to Notation.