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The Divergence Theorem and Sets of Finite Perimeter

The Divergence Theorem and Sets of Finite Perimeter PDF Author: Washek F. Pfeffer
Publisher: CRC Press
ISBN: 1466507195
Category : Mathematics
Languages : en
Pages : 261

Book Description
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 Divergence Theorem and Sets of Finite Perimeter

The Divergence Theorem and Sets of Finite Perimeter PDF Author: Washek F. Pfeffer
Publisher: CRC Press
ISBN: 1466507195
Category : Mathematics
Languages : en
Pages : 261

Book Description
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 Divergence Theorem and Sets of Finite Perimeter

The Divergence Theorem and Sets of Finite Perimeter PDF Author: Washek F. Pfeffer
Publisher: CRC Press
ISBN: 1466507217
Category : Mathematics
Languages : en
Pages : 259

Book Description
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 an

Sets of Finite Perimeter and Geometric Variational Problems

Sets of Finite Perimeter and Geometric Variational Problems PDF Author: Francesco Maggi
Publisher: Cambridge University Press
ISBN: 1107021030
Category : Mathematics
Languages : en
Pages : 475

Book Description
An engaging graduate-level introduction that bridges analysis and geometry. Suitable for self-study and a useful reference for researchers.

Geometric Harmonic Analysis I

Geometric Harmonic Analysis I PDF Author: Dorina Mitrea
Publisher: Springer Nature
ISBN: 3031059506
Category : Mathematics
Languages : en
Pages : 940

Book Description
This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. Volume I establishes a sharp version of the Divergence Theorem (aka Fundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundary trace is only assumed to exist in a nontangential pointwise sense.

Integral Operators in Potential Theory

Integral Operators in Potential Theory PDF Author: Josef Kral
Publisher: Springer
ISBN: 3540382887
Category : Mathematics
Languages : en
Pages : 175

Book Description


Measure Theory

Measure Theory PDF Author: D. H. Fremlin
Publisher: Torres Fremlin
ISBN: 0953812944
Category : Fourier analysis
Languages : en
Pages : 967

Book Description


Minimal Surfaces of Codimension One

Minimal Surfaces of Codimension One PDF Author: U. Massari
Publisher: Elsevier
ISBN: 0080872026
Category : Science
Languages : en
Pages : 259

Book Description
This book gives a unified presentation of different mathematical tools used to solve classical problems like Plateau's problem, Bernstein's problem, Dirichlet's problem for the Minimal Surface Equation and the Capillary problem.The fundamental idea is a quite elementary geometrical definition of codimension one surfaces. The isoperimetric property of the Euclidean balls, together with the modern theory of partial differential equations are used to solve the 19th Hilbert problem. Also included is a modern mathematical treatment of capillary problems.

Singularities in PDE and the Calculus of Variations

Singularities in PDE and the Calculus of Variations PDF Author: Stanley Alama
Publisher: American Mathematical Soc.
ISBN: 9780821873311
Category : Mathematics
Languages : en
Pages : 284

Book Description
This book contains papers presented at the "Workshop on Singularities in PDE and the Calculus of Variations" at the CRM in July 2006. The main theme of the meeting was the formation of geometrical singularities in PDE problems with a variational formulation. These equations typically arise in some applications (to physics, engineering, or biology, for example) and their resolution often requires a combination of methods coming from areas such as functional and harmonic analysis, differential geometry and geometric measure theory. Among the PDE problems discussed were: the Cahn-Hilliard model of phase transitions and domain walls; vortices in Ginzburg-Landau type models for superconductivity and superfluidity; the Ohna-Kawasaki model for di-block copolymers; models of image enhancement; and Monge-Ampere functions. The articles give a sampling of problems and methods in this diverse area of mathematics, which touches a large part of modern mathematics and its applications.

Calculus of Variations and Nonlinear Partial Differential Equations

Calculus of Variations and Nonlinear Partial Differential Equations PDF Author: Luigi Ambrosio
Publisher: Springer Science & Business Media
ISBN: 3540759131
Category : Mathematics
Languages : en
Pages : 213

Book Description
With a historical overview by Elvira Mascolo

Fluid Mechanics of Viscoelasticity

Fluid Mechanics of Viscoelasticity PDF Author: R.R. Huilgol
Publisher: Elsevier
ISBN: 0080531741
Category : Science
Languages : en
Pages : 508

Book Description
The areas of suspension mechanics, stability and computational rheology have exploded in scope and substance in the last decade. The present book is one of the first of a comprehensive nature to treat these topics in detail. The aim of the authors has been to highlight the major discoveries and to present a number of them in sufficient breadth and depth so that the novice can learn from the examples chosen, and the expert can use them as a reference when necessary.The first two chapters, grouped under the category General Principles, deal with the kinematics of continuous media and the balance laws of mechanics, including the existence of the stress tensor and extensions of the laws of vector analysis to domains bounded by fractal curves or surfaces. The third and fourth chapters, under the heading Constitutive Modelling, present the tools necessary to formulate constitutive equations from the continuum or the microstructural approach. The last three chapters, under the caption Analytical and Numerical Techniques, contain most of the important results in the domain of the fluid mechanics of viscoelasticity, and form the core of the book.A number of topics of interest have not yet been developed to a theoretical level from which applications can be made in a routine manner. However, the authors have included these topics to make the reader aware of the state of affairs so that research into these matters can be carried out. For example, the sections which deal with domains bounded by fractal curves or surfaces show that the existence of a stress tensor in such regions is still open to question. Similarly, the constitutive modelling of suspensions, especially at high volume concentrations, with the corresponding particle migration from high to low shear regions is still very sketchy.