The Divergence Theorem and Sets of Finite Perimeter
Features
- Provides new material on analysis, including geometric measure theory, bounded variation sets, and geometric topology
- Presents the important theorems from analysis, carefully developing concepts and machinery to rigorously validate the results
- Covers dyadic figures, sets of finite perimeter, and both bounded and unbounded vector fields
Summary
This book is devoted to a detailed development of the divergence theorem. The framework is that of Lebesgue integration — no generalized Riemann integrals of Henstock–Kurzweil variety are involved.
In Part I the divergence theorem is established by a combinatorial argument involving dyadic cubes. Only elementary properties of the Lebesgue integral and Hausdorff measures are used. The resulting integration by parts is sufficiently general for many applications. As an example, it is applied to removable singularities of Cauchy–Riemann, Laplace, and minimal surface equations.
The sets of finite perimeter are introduced in Part II. Both the geometric and analytic points of view are presented. The equivalence of these viewpoints is obtained via the functions of bounded variation. These functions are studied in a self-contained manner with no references to Sobolev’s spaces. The coarea theorem provides a link between the sets of finite perimeter and functions of bounded variation.
The general divergence theorem for bounded vector fields is proved in Part III. The proof consists of adapting the combinatorial argument of Part I to sets of finite perimeter. The unbounded vector fields and mean divergence are also discussed. The final chapter contains a characterization of the distributions that are equal to the flux of a continuous vector field.
Table of Contents
DYADIC FIGURES
Preliminaries
The setting
Topology
Measures
Hausdorff measures
Differentiable and Lipschitz maps
Divergence Theorem for Dyadic Figures
Differentiable vector fields
Dyadic partitions
Admissible maps
Convergence of dyadic figures
Removable Singularities
Distributions
Differential equations
Holomorphic functions
Harmonic functions
The minimal surface equation
Injective limits
SETS OF FINITE PERIMETER
Perimeter
Measure-theoretic concepts
Essential boundary
Vitali’s covering theorem
Density
Definition of perimeter
Line sections
BV Functions
Variation
Mollification
Vector valued measures
Weak convergence
Properties of BV functions
Approximation theorem
Coarea theorem
Bounded convex domains
Inequalities
Locally BV Sets
Dimension one
Besicovitch’s covering theorem
The reduced boundary
Blow-up
Perimeter and variation
Properties of BV sets
Approximating by figures
THE DIVERGENCE THEOREM
Bounded Vector Fields
Approximating from inside
Relative derivatives
The critical interior
The divergence theorem
Lipschitz domains
Unbounded Vector Fields
Minkowski contents
Controlled vector fields
Integration by parts
Mean Divergence
The derivative
The critical variation
Charges
Continuous vector fields
Localized topology
Locally convex spaces
Duality
The space BVc(Ω)
Streams
The Divergence Equation
Background
Solutions in Lp(Ω; Rn)
Continuous solutions
Bibliography
List of Symbols
Index
Editorial Reviews
"The intentions of the author connected with the entire monograph are best illustrated by a quotation from the introduction: ‘We divide the problem into three parts. (1) Extending the family of vector fields for which the divergence theorem holds on simple sets. (2) Extending the family of sets for which the divergence theorem holds for Lipschitz vector fields. (3) Proving the divergence theorem when the vector fields and sets are extended simultaneously.’ … The last chapter … [contain] results published for the first time in this century. The author starts these considerations with a nice presentation of the background of these problems."
—Ryszard J. Pawlak, Mathematical Reviews, April 2013
