The Divergence Theorem and Sets of Finite Perimeter

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