The Geometry of Domains in Space - Softcover

Inhaltsangabe

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.

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