The Divergence Theorem and Sets of Finite Perimeter (Hardback)

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.

"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

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 BV_c(Ω)

Streams

The Divergence Equation

Background

Solutions in L^p(Ω; Rⁿ)

Continuous solutions

Bibliography

List of Symbols

Index

The Divergence Theorem and Sets of Finite Perimeter

By Washek F. Pfeffer

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