Pfeffer The Divergence Theorem and Sets of Finite Perimeter

1. Auflage 2012
ISBN: 978-1-4665-0721-0
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 259 Seiten

Reihe: Chapman & Hall/CRC Pure and Applied Mathematics

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.

Zielgruppe

Mathematicians, mechanical/electrical/control engineers, and physicists; students in mathematics or engineering.

Autoren/Hrsg.

Pfeffer, Washek F.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Mathematische Analysis Vektoranalysis, Physikalische Felder
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik

Inhaltsverzeichnis

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(O)
Streams

The Divergence Equation
Background
Solutions in Lp(O; Rn)
Continuous solutions

Bibliography
List of Symbols
Index