Geometric Measure Theory - An Introduction

This book is intended for the researchers engaged in fields of mathematics and 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 calculus of variations, geometrical analysis and nonlinear partial differential equations. The contents of the book mainly consist of the Hausdoff measure and its applications, Lipschitz functions rectifiable sets, the area and coarea formulae, BV functions and sets of the finite perimeter, Varifolds theory, and current theory. The present text also contains many concrete examples illustrating how the basic ideas and powerful techniques in geometric measure theory are applied.

Chapter 1 Hausdorff 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
Chapter 2 Fine Properties of Functions and Sets and Their Applications
2.1 Lebesgue Points of Sobolev Functions
2.2 Self-Similar Sets
2.3 Federer's Reduction Principle
Chapter 3 Lipschitz Functions and Rectifiable Sets
3.1 Lipschitz Functions
3.2 Submanifolds of Rn+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 Rectifiability
Chapter 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 Formulae
Chapter 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
Chapter 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
Chapter 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
Chapter 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
Bibliography
Index