The divergence theorem and sets of finite perimeter
- Pfeffer, Washek F.
- Boca Raton : CRC Press, c2012.
- Physical description
- xv, 242 p. : ill. ; 25 cm.
- Monographs and textbooks in pure and applied mathematics ; 303.
QA433 .P493 2012
- Unknown QA433 .P493 2012
- Includes bibliographical references (p. 231-233) and index.
- 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(OMEGA) Streams The Divergence Equation Background Solutions in Lp(OMEGA-- Rn) Continuous solutions Bibliography List of Symbols Index.
- (source: Nielsen Book Data)
- Publisher's 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.
(source: Nielsen Book Data)
- Publication date
- Washek F. Pfeffer.
- Monographs and textbooks in pure and applied mathematics ; 303
- "A Chapman & Hall book."