The divergence theorem and sets of finite perimeter
 Author/Creator
 Pfeffer, Washek F.
 Language
 English.
 Imprint
 Boca Raton : CRC Press, c2012.
 Physical description
 xv, 242 p. : ill. ; 25 cm.
 Series
 Monographs and textbooks in pure and applied mathematics ; 303.
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QA433 .P493 2012

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QA433 .P493 2012
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Contents/Summary
 Bibliography
 Includes bibliographical references (p. 231233) and index.
 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 Measuretheoretic 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 Blowup 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 HenstockKurzweil 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 CauchyRiemann, 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 selfcontained 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)
Subjects
Bibliographic information
 Publication date
 2012
 Responsibility
 Washek F. Pfeffer.
 Series
 Monographs and textbooks in pure and applied mathematics ; 303
 Note
 "A Chapman & Hall book."
 ISBN
 9781466507197 (hardback)
 1466507195 (hardback)