Functional Spaces For The Theory Of Elliptic Partial Differential Equations

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Functional Spaces for the Theory of Elliptic Partial Differential Equations

Françoise Demengel,Gilbert Demengel
Functional Spaces for the Theory of Elliptic Partial Differential Equations Book PDF

Author : Françoise Demengel,Gilbert Demengel
Publisher : Springer Science & Business Media
Page : 480 pages
File Size : 45,7 Mb
Release : 2012-01-24
Category : Mathematics
ISBN : 9781447128076

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Functional Spaces for the Theory of Elliptic Partial Differential Equations by Françoise Demengel,Gilbert Demengel Pdf

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.

Variational Techniques for Elliptic Partial Differential Equations

Francisco J. Sayas,Thomas S. Brown,Matthew E. Hassell
Variational Techniques for Elliptic Partial Differential Equations Book PDF

Author : Francisco J. Sayas,Thomas S. Brown,Matthew E. Hassell
Publisher : CRC Press
Page : 492 pages
File Size : 43,9 Mb
Release : 2019-01-16
Category : Mathematics
ISBN : 9780429016202

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Variational Techniques for Elliptic Partial Differential Equations by Francisco J. Sayas,Thomas S. Brown,Matthew E. Hassell Pdf

Variational Techniques for Elliptic Partial Differential Equations, intended for graduate students studying applied math, analysis, and/or numerical analysis, provides the necessary tools to understand the structure and solvability of elliptic partial differential equations. Beginning with the necessary definitions and theorems from distribution theory, the book gradually builds the functional analytic framework for studying elliptic PDE using variational formulations. Rather than introducing all of the prerequisites in the first chapters, it is the introduction of new problems which motivates the development of the associated analytical tools. In this way the student who is encountering this material for the first time will be aware of exactly what theory is needed, and for which problems. Features A detailed and rigorous development of the theory of Sobolev spaces on Lipschitz domains, including the trace operator and the normal component of vector fields An integration of functional analysis concepts involving Hilbert spaces and the problems which can be solved with these concepts, rather than separating the two Introduction to the analytical tools needed for physical problems of interest like time-harmonic waves, Stokes and Darcy flow, surface differential equations, Maxwell cavity problems, etc. A variety of problems which serve to reinforce and expand upon the material in each chapter, including applications in fluid and solid mechanics

Functional Analysis, Sobolev Spaces and Partial Differential Equations

Haim Brezis
Functional Analysis Sobolev Spaces and Partial Differential Equations Book PDF

Author : Haim Brezis
Publisher : Springer Science & Business Media
Page : 600 pages
File Size : 49,9 Mb
Release : 2010-11-02
Category : Mathematics
ISBN : 9780387709147

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Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis Pdf

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.

Sobolev Spaces

Vladimir Maz'ya
Sobolev Spaces Book PDF

Author : Vladimir Maz'ya
Publisher : Springer Science & Business Media
Page : 882 pages
File Size : 49,7 Mb
Release : 2011-02-11
Category : Mathematics
ISBN : 9783642155642

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Sobolev Spaces by Vladimir Maz'ya Pdf

Sobolev spaces play an outstanding role in modern analysis, in particular, in the theory of partial differential equations and its applications in mathematical physics. They form an indispensable tool in approximation theory, spectral theory, differential geometry etc. The theory of these spaces is of interest in itself being a beautiful domain of mathematics. The present volume includes basics on Sobolev spaces, approximation and extension theorems, embedding and compactness theorems, their relations with isoperimetric and isocapacitary inequalities, capacities with applications to spectral theory of elliptic differential operators as well as pointwise inequalities for derivatives. The selection of topics is mainly influenced by the author’s involvement in their study, a considerable part of the text is a report on his work in the field. Part of this volume first appeared in German as three booklets of Teubner-Texte zur Mathematik (1979, 1980). In the Springer volume “Sobolev Spaces”, published in English in 1985, the material was expanded and revised. The present 2nd edition is enhanced by many recent results and it includes new applications to linear and nonlinear partial differential equations. New historical comments, five new chapters and a significantly augmented list of references aim to create a broader and modern view of the area.

Elliptic Differential Operators and Spectral Analysis

D. E. Edmunds,W.D. Evans
Elliptic Differential Operators and Spectral Analysis Book PDF

Author : D. E. Edmunds,W.D. Evans
Publisher : Springer
Page : 322 pages
File Size : 40,9 Mb
Release : 2018-11-20
Category : Mathematics
ISBN : 9783030021252

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Elliptic Differential Operators and Spectral Analysis by D. E. Edmunds,W.D. Evans Pdf

This book deals with elliptic differential equations, providing the analytic background necessary for the treatment of associated spectral questions, and covering important topics previously scattered throughout the literature. Starting with the basics of elliptic operators and their naturally associated function spaces, the authors then proceed to cover various related topics of current and continuing importance. Particular attention is given to the characterisation of self-adjoint extensions of symmetric operators acting in a Hilbert space and, for elliptic operators, the realisation of such extensions in terms of boundary conditions. A good deal of material not previously available in book form, such as the treatment of the Schauder estimates, is included. Requiring only basic knowledge of measure theory and functional analysis, the book is accessible to graduate students and will be of interest to all researchers in partial differential equations. The reader will value its self-contained, thorough and unified presentation of the modern theory of elliptic operators.

Sobolev Spaces

Vladimir Maz'ya
Sobolev Spaces Book PDF

Author : Vladimir Maz'ya
Publisher : Springer
Page : 506 pages
File Size : 41,9 Mb
Release : 2013-12-21
Category : Mathematics
ISBN : 9783662099223

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Sobolev Spaces by Vladimir Maz'ya Pdf

The Sobolev spaces, i. e. the classes of functions with derivatives in L , occupy p an outstanding place in analysis. During the last two decades a substantial contribution to the study of these spaces has been made; so now solutions to many important problems connected with them are known. In the present monograph we consider various aspects of Sobolev space theory. Attention is paid mainly to the so called imbedding theorems. Such theorems, originally established by S. L. Sobolev in the 1930s, proved to be a useful tool in functional analysis and in the theory of linear and nonlinear par tial differential equations. We list some questions considered in this book. 1. What are the requirements on the measure f1, for the inequality q

Lectures on Elliptic Partial Differential Equations

Luigi Ambrosio,Alessandro Carlotto,Annalisa Massaccesi
Lectures on Elliptic Partial Differential Equations Book PDF

Author : Luigi Ambrosio,Alessandro Carlotto,Annalisa Massaccesi
Publisher : Springer
Page : 230 pages
File Size : 48,7 Mb
Release : 2019-01-10
Category : Mathematics
ISBN : 9788876426513

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Lectures on Elliptic Partial Differential Equations by Luigi Ambrosio,Alessandro Carlotto,Annalisa Massaccesi Pdf

The book originates from the Elliptic PDE course given by the first author at the Scuola Normale Superiore in recent years. It covers the most classical aspects of the theory of Elliptic Partial Differential Equations and Calculus of Variations, including also more recent developments on partial regularity for systems and the theory of viscosity solutions.

Elliptic Partial Differential Equations

Vitaly Volpert
Elliptic Partial Differential Equations Book PDF

Author : Vitaly Volpert
Publisher : Springer Science & Business Media
Page : 649 pages
File Size : 54,6 Mb
Release : 2011-03-03
Category : Mathematics
ISBN : 9783034605373

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Elliptic Partial Differential Equations by Vitaly Volpert Pdf

The theory of elliptic partial differential equations has undergone an important development over the last two centuries. Together with electrostatics, heat and mass diffusion, hydrodynamics and many other applications, it has become one of the most richly enhanced fields of mathematics. This monograph undertakes a systematic presentation of the theory of general elliptic operators. The author discusses a priori estimates, normal solvability, the Fredholm property, the index of an elliptic operator, operators with a parameter, and nonlinear Fredholm operators. Particular attention is paid to elliptic problems in unbounded domains which have not yet been sufficiently treated in the literature and which require some special approaches. The book also contains an analysis of non-Fredholm operators and discrete operators as well as extensive historical and bibliographical comments . The selected topics and the author's level of discourse will make this book a most useful resource for researchers and graduate students working in the broad field of partial differential equations and applications.

Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains

Mikhail S. Agranovich
Sobolev Spaces Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains Book PDF

Author : Mikhail S. Agranovich
Publisher : Springer
Page : 331 pages
File Size : 46,8 Mb
Release : 2015-05-06
Category : Mathematics
ISBN : 9783319146485

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Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains by Mikhail S. Agranovich Pdf

This book, which is based on several courses of lectures given by the author at the Independent University of Moscow, is devoted to Sobolev-type spaces and boundary value problems for linear elliptic partial differential equations. Its main focus is on problems in non-smooth (Lipschitz) domains for strongly elliptic systems. The author, who is a prominent expert in the theory of linear partial differential equations, spectral theory and pseudodifferential operators, has included his own very recent findings in the present book. The book is well suited as a modern graduate textbook, utilizing a thorough and clear format that strikes a good balance between the choice of material and the style of exposition. It can be used both as an introduction to recent advances in elliptic equations and boundary value problems and as a valuable survey and reference work. It also includes a good deal of new and extremely useful material not available in standard textbooks to date. Graduate and post-graduate students, as well as specialists working in the fields of partial differential equations, functional analysis, operator theory and mathematical physics will find this book particularly valuable.

Elliptic Differential Equations

Wolfgang Hackbusch
Elliptic Differential Equations Book PDF

Author : Wolfgang Hackbusch
Publisher : Springer
Page : 455 pages
File Size : 51,6 Mb
Release : 2017-06-01
Category : Mathematics
ISBN : 9783662549612

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Elliptic Differential Equations by Wolfgang Hackbusch Pdf

This book simultaneously presents the theory and the numerical treatment of elliptic boundary value problems, since an understanding of the theory is necessary for the numerical analysis of the discretisation. It first discusses the Laplace equation and its finite difference discretisation before addressing the general linear differential equation of second order. The variational formulation together with the necessary background from functional analysis provides the basis for the Galerkin and finite-element methods, which are explored in detail. A more advanced chapter leads the reader to the theory of regularity. Individual chapters are devoted to singularly perturbed as well as to elliptic eigenvalue problems. The book also presents the Stokes problem and its discretisation as an example of a saddle-point problem taking into account its relevance to applications in fluid dynamics.

Partial Differential Equations and Functional Analysis

J. Cea,D. Chenais,Giuseppe Geymonat,J.L. Lions
Partial Differential Equations and Functional Analysis Book PDF

Author : J. Cea,D. Chenais,Giuseppe Geymonat,J.L. Lions
Publisher : Springer Science & Business Media
Page : 288 pages
File Size : 44,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461224365

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Partial Differential Equations and Functional Analysis by J. Cea,D. Chenais,Giuseppe Geymonat,J.L. Lions Pdf

Pierre Grisvard, one of the most distinguished French mathematicians, died on April 22, 1994. A Conference was held in November 1994 out of which grew the invited articles contained in this volume. All of the papers are related to functional analysis applied to partial differential equations, which was Grisvard's specialty. Indeed his knowledge of this area was extremely broad. He began his career as one of the very first students of Jacques Louis Lions, and in 1965, he presented his "State Thesis" on interpolation spaces, using in particular, spectral theory for linear operators in Banach spaces. After 1970, he became a specialist in the study of optimal regularity for par tial differential equations with boundary conditions. He studied singulari ties coming from coefficients, boundary conditions, and mainly non-smooth domains, and left a legacy of precise results which have been published in journals and books. Pierre Grisvard spent most of his career as a full professor at the University of Nice, where he started in 1967. For shorter or longer periods, he visited several foreign countries, and collaborated with some of the most famous mathematicians in his field. He was also an excellent organizer and directed a large number of Ph.D. students. Finally, this volume contains a bibliography of Grisvard's works as well as one paper which he wrote and which has not been published before.

Partial Differential Equations 2

Friedrich Sauvigny
Partial Differential Equations 2 Book PDF

Author : Friedrich Sauvigny
Publisher : Springer Science & Business Media
Page : 401 pages
File Size : 46,6 Mb
Release : 2006-10-11
Category : Mathematics
ISBN : 9783540344629

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Partial Differential Equations 2 by Friedrich Sauvigny Pdf

This encyclopedic work covers the whole area of Partial Differential Equations - of the elliptic, parabolic, and hyperbolic type - in two and several variables. Emphasis is placed on the connection of PDEs and complex variable methods. This second volume addresses Solvability of operator equations in Banach spaces; Linear operators in Hilbert spaces and spectral theory; Schauder's theory of linear elliptic differential equations; Weak solutions of differential equations; Nonlinear partial differential equations and characteristics; Nonlinear elliptic systems with differential-geometric applications. While partial differential equations are solved via integral representations in the preceding volume, this volume uses functional analytic solution methods.

Stable Solutions of Elliptic Partial Differential Equations

Louis Dupaigne
Stable Solutions of Elliptic Partial Differential Equations Book PDF

Author : Louis Dupaigne
Publisher : CRC Press
Page : 337 pages
File Size : 45,6 Mb
Release : 2011-03-15
Category : Mathematics
ISBN : 9781420066548

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Stable Solutions of Elliptic Partial Differential Equations by Louis Dupaigne Pdf

Stable solutions are ubiquitous in differential equations. They represent meaningful solutions from a physical point of view and appear in many applications, including mathematical physics (combustion, phase transition theory) and geometry (minimal surfaces). Stable Solutions of Elliptic Partial Differential Equations offers a self-contained presentation of the notion of stability in elliptic partial differential equations (PDEs). The central questions of regularity and classification of stable solutions are treated at length. Specialists will find a summary of the most recent developments of the theory, such as nonlocal and higher-order equations. For beginners, the book walks you through the fine versions of the maximum principle, the standard regularity theory for linear elliptic equations, and the fundamental functional inequalities commonly used in this field. The text also includes two additional topics: the inverse-square potential and some background material on submanifolds of Euclidean space.

Elliptic Functional Differential Equations and Applications

Alexander L. Skubachevskii
Elliptic Functional Differential Equations and Applications Book PDF

Author : Alexander L. Skubachevskii
Publisher : Birkhäuser
Page : 298 pages
File Size : 43,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034890335

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Elliptic Functional Differential Equations and Applications by Alexander L. Skubachevskii Pdf

Boundary value problems for elliptic differential-difference equations have some astonishing properties. For example, unlike elliptic differential equations, the smoothness of the generalized solutions can be broken in a bounded domain and is preserved only in some subdomains. The symbol of a self-adjoint semibounded functional differential operator can change its sign. The purpose of this book is to present for the first time general results concerning solvability and spectrum of these problems, a priori estimates and smoothness of solutions. The approach is based on the properties of elliptic operators and difference operators in Sobolev spaces. The most important features distinguishing this work are applications to different fields of science. The methods in this book are used to obtain new results regarding the solvability of nonlocal elliptic boundary value problems and the existence of Feller semigroups for multidimensional diffusion processes. Moreover, applications to control theory and aircraft and rocket technology are given. The theory is illustrated with numerous figures and examples. The book is addresssed to graduate students and researchers in partial differential equations and functional differential equations. It will also be of use to engineers in control theory and elasticity theory.

Partial Differential Equations

Michael Shearer,Rachel Levy
Partial Differential Equations Book PDF

Author : Michael Shearer,Rachel Levy
Publisher : Princeton University Press
Page : 286 pages
File Size : 53,8 Mb
Release : 2015-03-01
Category : Mathematics
ISBN : 9780691161297

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Partial Differential Equations by Michael Shearer,Rachel Levy Pdf

An accessible yet rigorous introduction to partial differential equations This textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations (PDEs). It presents a rigorous and clear explanation of the more elementary theoretical aspects of PDEs, while also drawing connections to deeper analysis and applications. The book serves as a needed bridge between basic undergraduate texts and more advanced books that require a significant background in functional analysis. Topics include first order equations and the method of characteristics, second order linear equations, wave and heat equations, Laplace and Poisson equations, and separation of variables. The book also covers fundamental solutions, Green's functions and distributions, beginning functional analysis applied to elliptic PDEs, traveling wave solutions of selected parabolic PDEs, and scalar conservation laws and systems of hyperbolic PDEs. Provides an accessible yet rigorous introduction to partial differential equations Draws connections to advanced topics in analysis Covers applications to continuum mechanics An electronic solutions manual is available only to professors An online illustration package is available to professors