Distributions, Sobolev Spaces, Elliptic Equations
Distributions, Sobolev Spaces, Elliptic Equations

Author: Dorothee Haroske

Publisher: European Mathematical Society

Published: 2007

Total Pages: 312

ISBN-13: 9783037190425

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It is the main aim of this book to develop at an accessible, moderate level an $L_2$ theory for elliptic differential operators of second order on bounded smooth domains in Euclidean n-space, including a priori estimates for boundary-value problems in terms of (fractional) Sobolev spaces on domains and on their boundaries, together with a related spectral theory. The presentation is preceded by an introduction to the classical theory for the Laplace-Poisson equation, and some chapters provide required ingredients such as the theory of distributions, Sobolev spaces and the spectral theory in Hilbert spaces. The book grew out of two-semester courses the authors have given several times over a period of ten years at the Friedrich Schiller University of Jena. It is addressed to graduate students and mathematicians who have a working knowledge of calculus, measure theory and the basic elements of functional analysis (as usually covered by undergraduate courses) and who are seeking an accessible introduction to some aspects of the theory of function spaces and its applications to elliptic equations.

Functional Spaces for the Theory of Elliptic Partial Differential Equations
Functional Spaces for the Theory of Elliptic Partial Differential Equations

Author: Françoise Demengel

Publisher: Springer Science & Business Media

Published: 2012-01-24

Total Pages: 480

ISBN-13: 1447128079

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The theory of elliptic boundary problems is fundamental in analysis and the role of spaces of weakly differentiable functions (also called Sobolev spaces) is essential in this theory as a tool for analysing the regularity of the solutions. This book offers on the one hand a complete theory of Sobolev spaces, which are of fundamental importance for elliptic linear and non-linear differential equations, and explains on the other hand how the abstract methods of convex analysis can be combined with this theory to produce existence results for the solutions of non-linear elliptic boundary problems. The book also considers other kinds of functional spaces which are useful for treating variational problems such as the minimal surface problem. The main purpose of the book is to provide a tool for graduate and postgraduate students interested in partial differential equations, as well as a useful reference for researchers active in the field. Prerequisites include a knowledge of classical analysis, differential calculus, Banach and Hilbert spaces, integration and the related standard functional spaces, as well as the Fourier transformation on the Schwartz space. There are complete and detailed proofs of almost all the results announced and, in some cases, more than one proof is provided in order to highlight different features of the result. Each chapter concludes with a range of exercises of varying levels of difficulty, with hints to solutions provided for many of them.

Distributions
Distributions

Author: Pulin Kumar Bhattacharyya

Publisher: Walter de Gruyter

Published: 2012-05-29

Total Pages: 871

ISBN-13: 3110269295

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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.

Functional Analysis, Sobolev Spaces and Partial Differential Equations
Functional Analysis, Sobolev Spaces and Partial Differential Equations

Author: Haim Brezis

Publisher: Springer Science & Business Media

Published: 2010-11-02

Total Pages: 600

ISBN-13: 0387709142

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This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.

Theory of Function Spaces IV
Theory of Function Spaces IV

Author: Hans Triebel

Publisher: Springer Nature

Published: 2020-01-23

Total Pages: 167

ISBN-13: 3030358917

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This book is the continuation of the "Theory of Function Spaces" trilogy, published by the same author in this series and now part of classic literature in the area of function spaces. It can be regarded as a supplement to these volumes and as an accompanying book to the textbook by D.D. Haroske and the author "Distributions, Sobolev spaces, elliptic equations".

Distribution Theory Applied to Differential Equations
Distribution Theory Applied to Differential Equations

Author: Adina Chirilă

Publisher: Springer Nature

Published: 2021-02-08

Total Pages: 277

ISBN-13: 3030671593

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This book presents important contributions to modern theories concerning the distribution theory applied to convex analysis (convex functions, functions of lower semicontinuity, the subdifferential of a convex function). The authors prove several basic results in distribution theory and present ordinary differential equations and partial differential equations by providing generalized solutions. In addition, the book deals with Sobolev spaces, which presents aspects related to variation problems, such as the Stokes system, the elasticity system and the plate equation. The authors also include approximate formulations of variation problems, such as the Galerkin method or the finite element method. The book is accessible to all scientists, and it is especially useful for those who use mathematics to solve engineering and physics problems. The authors have avoided concepts and results contained in other books in order to keep the book comprehensive. Furthermore, they do not present concrete simplified models and pay maximal attention to scientific rigor.

Theory of Distributions
Theory of Distributions

Author: Svetlin G. Georgiev

Publisher: Springer Nature

Published: 2021-08-21

Total Pages: 270

ISBN-13: 3030812650

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This book explains many fundamental ideas on the theory of distributions. The theory of partial differential equations is one of the synthetic branches of analysis that combines ideas and methods from different fields of mathematics, ranging from functional analysis and harmonic analysis to differential geometry and topology. This presents specific difficulties to those studying this field. This second edition, which consists of 10 chapters, is suitable for upper undergraduate/graduate students and mathematicians seeking an accessible introduction to some aspects of the theory of distributions. It can also be used for one-semester course.

Fractional Sobolev Spaces and Inequalities
Fractional Sobolev Spaces and Inequalities

Author: D. E. Edmunds

Publisher: Cambridge University Press

Published: 2022-10-13

Total Pages: 170

ISBN-13: 1009254642

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The fractional Sobolev spaces studied in the book were introduced in the 1950s by Aronszajn, Gagliardo and Slobodeckij in an attempt to fill the gaps between the classical Sobolev spaces. They provide a natural home for solutions of a vast, and rapidly growing, number of questions involving differential equations and non-local effects, ranging from financial modelling to ultra-relativistic quantum mechanics, emphasising the need to be familiar with their fundamental properties and associated techniques. Following an account of the most basic properties of the fractional spaces, two celebrated inequalities, those of Hardy and Rellich, are discussed, first in classical format (for which a survey of the very extensive known results is given), and then in fractional versions. This book will be an Ideal resource for researchers and graduate students working on differential operators and boundary value problems.

Fundamentals of Applied Functional Analysis
Fundamentals of Applied Functional Analysis

Author: Dragisa Mitrovic

Publisher: CRC Press

Published: 1997-11-12

Total Pages: 416

ISBN-13: 9780582246942

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This volume provides an introduction to modern concepts of linear and nonlinear functional analysis. Its purpose is also to provide an insight into the variety of deeply interlaced mathematical tools applied in the study of nonlinear problems.