Geometric Measure Theory --- An introduction

Co-published by Science Press & International Press

Price: $55

ISBN: 1-57146-125-6 / ISBN: 7-03-010271-1

Binding: Hardcover

Page Number: 247

Year Published: 2002

Since the publication of the seminal work of H. Federer which gives a rather complete and comprehensive discussion on the subject, the geometric measure theory has developed in the last three decades into an even more cohesive body of basic knowledge with an ample structure of its own, established strong ties with many other subject areas of mathematics and made numerous new striking applications. The present book is intended for the researchers in other fields of mathematics as well as graduate students for a quick overview on the subject of the geometric measure theory with emphases on various basic ideas, techniques and their applications in problems arising in the calculus of variations, geometrical analysis and nonlinear partial differential equations.

This graduate-level treatment of Geometric Measure Theory illustrates with concrete examples and emphasizes basic ideas and techniques with their applications to the calculus of variations, geometrical analysis, and non-linear PDEs. The book, in addition to a full index and bibliography, include eight main chapters:

1 Hausdoff Measure

1.1 Preliminaries, Definitions and Properties

1.2 Isodiametric Inequality and Hn = Ln

1.3 Densities

1.4 Some Further Extensions Related to Hausdorff Measures

2 Fine Properties of Functions and Sets and Their Applications

2.1 Lebesgbue Points of Sobolev Functions

2.2 Self-Similar Sets

2.3 Federer’s Reduction Principle

3 Lipschitz Functions and Rectifiable Sets

3.1 Lipschitz Functions

3.2 Submanifolds of R n+k

3.3 Countably n-Rectifiable Sets

3.4 Weak Tangent Space Property, Measures in Cones and Rectifiability

3.5 Density and Rectifiability

3.6 Orthogonal Projections and Recifiability

4 The Area and Co-area Formulae

4.1 Area Formula and Its Proof

4.2 Co-area Formula

4.3 Some Extensions and Remarks

4.4 The First and Second Variation Formula

5 BV Functions and Sets of Finite Perimeter

5.1 Introduction and Definitions

5.2 Properties

5.3 Sobolev and Isoperimetric Inequalities

5.4 The Co-area Formula for BV Functions

5.5 The Reduced Boundary

5.6 Further Properties and Results Relative to BV Functions

6 Theory of Varifolds

6.1 Measures of Oscillation

6.2 Basic Definitions and the First Variation

6.3 Monotonicity Formula and Isoperimetric Inequality

6.4 Rectifiability Theorem and Tangent Cones

6.5 The Regularity Theory

7 Theory of Currents

7.1 Forms and Currents

7.2 Mapping Currents

7.3 Integral Rectifiable Currents

7.4 Deformation Theorem

7.5 Rectifiability of Currents

7.6 Compactness Theorem

8 Mass Minimizing Currents

8.1 Properties of Area Minimizing Currents

8.2 Excess and Height Bound

8.3 Excess Decay Lemmas and Regularity Theory