Distributions : : Generalized Functions with Applications in Sobolev Spaces / / Pulin Kumar Bhattacharyya.

• Superior document: Title is part of eBook package: De Gruyter DGBA Backlist Complete English Language 2000-2014 PART1
• VerfasserIn: Bhattacharyya, Pulin Kumar,
• Place / Publishing House: Berlin ;, Boston : : De Gruyter, , [2012]; ©2012
• Year of Publication: 2012
• Language: English
• Series: De Gruyter Textbook
• Physical Description: 1 online resource (834 p.)

This book grew out of a course taught in the Department of Mathematics, Indian Institute of Technology, Delhi, which was tailored to the needs of the applied community of mathematicians, engineers, physicists etc., who were interested in studying the problems of mathematical physics in general and their approximate solutions on computer in particular. Almost all topics which will be essential for the study of Sobolev spaces and their applications in the elliptic boundary value problems and their finite element approximations are presented. Also many additional topics of interests for specific applied disciplines and engineering, for example, elementary solutions, derivatives of discontinuous functions of several variables, delta-convergent sequences of functions, Fourier series of distributions, convolution system of equations etc. have been included along with many interesting examples.

Table of Contents

Frontmatter -- Preface -- Contents -- How to use this book in courses -- Acknowledgment -- Notation -- Chapter 1. Schwartz distributions -- Chapter 2. Differentiation of distributions and application of distributional derivatives -- Chapter 3. Derivatives of piecewise smooth functions, Green’s formula, elementary solutions, applications to Sobolev spaces -- Chapter 4. Additional properties of Dʹ(Ω) -- Chapter 5. Local properties, restrictions, unification principle, space ℰʹ(ℝn) of distributions with compact support -- Chapter 6. Convolution of distributions -- Chapter 7. Fourier transforms of functions of L1(ℝn) and S(ℝn) -- Chapter 8. Fourier transforms of distributions and Sobolev spaces of arbitrary order HS(ℝn) -- 8.1 Motivation for a possible definition of the Fourier transform of a distribution -- 8.2 Space Sʹ (Rn) of tempered distributions -- 8.3 Fourier transform of tempered distributions -- 8.4 Fourier transform of distributions with compact support -- 8.5 Fourier transform of convolution of distributions -- 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions -- 8.7 Fourier transform methods for differential equations and elementary solutions in Sʹ(ℝn) -- 8.8 Laplace transform of distributions on ℝ -- 8.9 Applications -- 8.10 Sobolev spaces on Ω ≠ Rn revisited -- 8.11 Compactness results in Sobolev spaces -- 8.12 Sobolev’s imbedding results -- 8.13 Sobolev spaces Hs.(Γ), Ws;p(Γ) on a manifold boundary Γ -- 8.14 Trace results in Sobolev spaces on Ω⊊ℝn -- Chapter 9. Vector-valued distributions -- Appendix A. Functional analysis (basic results) -- Appendix B. Lp-spaces -- Appendix C. Open cover and partition of unity -- Appendix D. Boundary geometry -- Bibliography -- Index