The Geometry of Domains in Space (Birkhäuser Advanced Texts Basler Lehrbücher)

Cover......Page 1
Birkhauser Advanced Texts......Page 2
Title......Page 3
ISBN 0-8176-4097-5  ISBN 3-7643-4097-5......Page 4
Dedication......Page 5
Contents......Page 7
Preface......Page 9
1.1 Smooth Functions......Page 11
1.2 The Concept of Defining Function......Page 18
1.3 Measure Theory......Page 25
2.1 The Tangent Bundle and Normal Bundle of the Boundary......Page 37
2.2 The Second Fundamental Form and Curvature......Page 44
2.3 Surfaces with Constant Mean Curvature......Page 55
3.1 The Caratheodory Construction......Page 66
3.2 Rectifiability......Page 75
3.3 Minkowski Content......Page 83
3.4 A Space-Filling Curve......Page 90
3.5 Covering Lemmas......Page 92
3.6 Functions of Bounded Variation......Page 109
3.7 Domains with Finite Perimeter......Page 116
3.8 The Area Formula......Page 132
3.9 The Co-Area Formula......Page 145
4.1 Basic Definitions and Results......Page 152
4.2 Restriction and Trace Theorellls for Sobolev Spaces......Page 157
4.3 Domain Extension Theorems for Sobolev Spaces......Page 162
5.1 Sard\'s Theorem......Page 165
5.2 Extension Theorems......Page 170
5.3 Proof of the Whitney Extension Theorem......Page 175
5.4 Application of the Whitney Extension Theorem......Page 187
5.5 Multidimensional Versions of the Fundamental Theorem of Calculus......Page 191
6.1 The Classical Notion of Convex.ity......Page 198
6.2 Other Characterizations of Convexity......Page 208
6.3 Exhaustion Functions......Page 218
6.4 Convexity of Order k......Page 224
7.1 Basic Properties......Page 229
7.2 The Isodiametric, Isoperimetric, and Brunn-Minkowski Inequalities......Page 237
7.3 Equality in the Isoperimetric Inequality......Page 249
8.1 Quasiconformal Mappings......Page 253
TheoreIn 8.1.4 (Liouville [I))......Page 259
8.2 Weyl\'s Theorem on Eigenvalue Asymptotics of a Domain in Space......Page 265
A. Hilbert Spaces......Page 268
B. MaxhnUID-MinilDUID Methods......Page 269
TheorelD 8.2.7 (DolDain Monotonicity)......Page 273
C. Eigenvalue Problems on Rectangular Parallelepipeds......Page 274
D. Eigenvalue Problems on Arbitrary Smooth Domains......Page 276
E. Proofs of the Results in B......Page 277
A.1 Metrics on the Collection of Subsets of Euclidean Space......Page 280
A.2 The Constants Associated to Euclidean Space......Page 286
Guide to Notation......Page 291
Bibliography......Page 294
Index......Page 308
Back Cover......Page 313