The Geometry of Domains in Space: (Birkhauser Advanced Texts / Basler Lehrbucher)

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.

Table of Contents:
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.

Review :
"This monograph collects a number of concepts, techniques and results of geometrical nature, centered around the concept of domain, and which are widely used by analysts. This includes the notion of defining functions for a bounded domain, techniques related to the smoothness of the boundary, some measure theory, including rectifiable sets, Minkowski content, covering lemmas, functions with bounded variation, and the area and co-area formula. Then comes a study of the restriction, trace and extension of functions belonging to a Sobolev space. One chapter is devoted to Sard's theorem and its application to the Whitney extension theorem, and another one to convexity and some of its generalizations. Steiner symmetrization is then treated, with its applications to isoperimetric inequalities. The last chapter deals with some questions related to complex analysis, namely quasiconformal mappings and Weyl's theorems on the asymptotic expression of eigenvalues. Two appendices deal with some metrics on the collection of subsets of a Euclidean space and some basic constants associated to those spaces. A short bibliography and an index complete this book, which is clearly written and makes an interesting link between analysis and geometry." --Zentralblatt Math "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