Variational Techniques For Elliptic Partial Differential Equations

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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 : 54,8 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

Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations

Vicentiu D. Radulescu,Vicenţiu D. Rădulescu
Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations Book PDF

Author : Vicentiu D. Radulescu,Vicenţiu D. Rădulescu
Publisher : Hindawi Publishing Corporation
Page : 205 pages
File Size : 52,8 Mb
Release : 2008
Category : Differential equations, Elliptic
ISBN : 9789774540394

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Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations by Vicentiu D. Radulescu,Vicenţiu D. Rădulescu Pdf

This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics, and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis. Concentrating on single-valued or multivalued elliptic equations with nonlinearities of various types, the aim of this volume is to obtain sharp existence or nonexistence results, as well as decay rates for general classes of solutions. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including bifurcation, stability, asymptotic analysis, and optimal regularity of solutions. The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. A systematic description of the most relevant singular phenomena described in this volume includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear phenomena described by elliptic partial differential equations.

Variational Methods for Boundary Value Problems for Systems of Elliptic Equations

M. A. Lavrent’ev
Variational Methods for Boundary Value Problems for Systems of Elliptic Equations Book PDF

Author : M. A. Lavrent’ev
Publisher : Courier Dover Publications
Page : 160 pages
File Size : 43,6 Mb
Release : 2016-01-14
Category : Mathematics
ISBN : 9780486160283

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Variational Methods for Boundary Value Problems for Systems of Elliptic Equations by M. A. Lavrent’ev Pdf

Famous monograph by a distinguished mathematician presents an innovative approach to classical boundary value problems. The treatment employs the basic scheme first suggested by Hilbert and developed by Tonnelli. 1963 edition.

Elliptic Differential Equations

Wolfgang Hackbusch
Elliptic Differential Equations Book PDF

Author : Wolfgang Hackbusch
Publisher : Springer
Page : 455 pages
File Size : 47,7 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.

Elliptic Partial Differential Equations

Qing Han,Fanghua Lin
Elliptic Partial Differential Equations Book PDF

Author : Qing Han,Fanghua Lin
Publisher : American Mathematical Soc.
Page : 161 pages
File Size : 45,5 Mb
Release : 2011
Category : Mathematics
ISBN : 9780821853139

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Elliptic Partial Differential Equations by Qing Han,Fanghua Lin Pdf

This volume is based on PDE courses given by the authors at the Courant Institute and at the University of Notre Dame, Indiana. Presented are basic methods for obtaining various a priori estimates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear; however, the presented methods also apply to nonlinear problems.

Elliptic Differential Equations

W. Hackbusch
Elliptic Differential Equations Book PDF

Author : W. Hackbusch
Publisher : Springer Science & Business Media
Page : 334 pages
File Size : 54,6 Mb
Release : 1992
Category : Language Arts & Disciplines
ISBN : 354054822X

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Elliptic Differential Equations by W. Hackbusch Pdf

Derived from a lecture series for college mathematics students, introduces the methods of dealing with elliptical boundary-value problems--both the theory and the numerical analysis. Includes exercises. Translated and somewhat expanded from the 1987 German version. Annotation copyright by Book News, Inc., Portland, OR

Semilinear Elliptic Equations for Beginners

Marino Badiale,Enrico Serra
Semilinear Elliptic Equations for Beginners Book PDF

Author : Marino Badiale,Enrico Serra
Publisher : Springer Science & Business Media
Page : 204 pages
File Size : 52,5 Mb
Release : 2010-12-07
Category : Mathematics
ISBN : 9780857292278

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Semilinear Elliptic Equations for Beginners by Marino Badiale,Enrico Serra Pdf

Semilinear elliptic equations are of fundamental importance for the study of geometry, physics, mechanics, engineering and life sciences. The variational approach to these equations has experienced spectacular success in recent years, reaching a high level of complexity and refinement, with a multitude of applications. Additionally, some of the simplest variational methods are evolving as classical tools in the field of nonlinear differential equations. This book is an introduction to variational methods and their applications to semilinear elliptic problems. Providing a comprehensive overview on the subject, this book will support both student and teacher engaged in a first course in nonlinear elliptic equations. The material is introduced gradually, and in some cases redundancy is added to stress the fundamental steps in theory-building. Topics include differential calculus for functionals, linear theory, and existence theorems by minimization techniques and min-max procedures. Requiring a basic knowledge of Analysis, Functional Analysis and the most common function spaces, such as Lebesgue and Sobolev spaces, this book will be of primary use to graduate students based in the field of nonlinear partial differential equations. It will also serve as valuable reading for final year undergraduates seeking to learn about basic working tools from variational methods and the management of certain types of nonlinear problems.

The Numerical Solution of Elliptic Equations

Garrett Birkhoff
The Numerical Solution of Elliptic Equations Book PDF

Author : Garrett Birkhoff
Publisher : SIAM
Page : 93 pages
File Size : 44,5 Mb
Release : 1971-01-01
Category : Mathematics
ISBN : 9780898710014

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The Numerical Solution of Elliptic Equations by Garrett Birkhoff Pdf

A concise survey of the current state of knowledge in 1972 about solving elliptic boundary-value eigenvalue problems with the help of a computer. This volume provides a case study in scientific computing?the art of utilizing physical intuition, mathematical theorems and algorithms, and modern computer technology to construct and explore realistic models of problems arising in the natural sciences and engineering.

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 : 44,6 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 Methods

Michael Struwe
Variational Methods Book PDF

Author : Michael Struwe
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 45,8 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783662041949

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Variational Methods by Michael Struwe Pdf

Hilberts talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateaus problem by Douglas and Rad. This third edition gives a concise introduction to variational methods and presents an overview of areas of current research in the field, plus a survey on new developments.

Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations

Mario Girardi
Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations Book PDF

Author : Mario Girardi
Publisher : Unknown
Page : 208 pages
File Size : 42,6 Mb
Release : 1992
Category : Mathematics
ISBN : UVA:X002085924

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Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations by Mario Girardi Pdf

This research note gives a comprehensive account of the use of variational methods in the study of Hamiltonian systems and elliptic equations.

Direct Methods in the Theory of Elliptic Equations

Jindrich Necas
Direct Methods in the Theory of Elliptic Equations Book PDF

Author : Jindrich Necas
Publisher : Springer Science & Business Media
Page : 384 pages
File Size : 51,6 Mb
Release : 2011-10-06
Category : Mathematics
ISBN : 9783642104558

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Direct Methods in the Theory of Elliptic Equations by Jindrich Necas Pdf

Nečas’ book Direct Methods in the Theory of Elliptic Equations, published 1967 in French, has become a standard reference for the mathematical theory of linear elliptic equations and systems. This English edition, translated by G. Tronel and A. Kufner, presents Nečas’ work essentially in the form it was published in 1967. It gives a timeless and in some sense definitive treatment of a number issues in variational methods for elliptic systems and higher order equations. The text is recommended to graduate students of partial differential equations, postdoctoral associates in Analysis, and scientists working with linear elliptic systems. In fact, any researcher using the theory of elliptic systems will benefit from having the book in his library. The volume gives a self-contained presentation of the elliptic theory based on the "direct method", also known as the variational method. Due to its universality and close connections to numerical approximations, the variational method has become one of the most important approaches to the elliptic theory. The method does not rely on the maximum principle or other special properties of the scalar second order elliptic equations, and it is ideally suited for handling systems of equations of arbitrary order. The prototypical examples of equations covered by the theory are, in addition to the standard Laplace equation, Lame’s system of linear elasticity and the biharmonic equation (both with variable coefficients, of course). General ellipticity conditions are discussed and most of the natural boundary condition is covered. The necessary foundations of the function space theory are explained along the way, in an arguably optimal manner. The standard boundary regularity requirement on the domains is the Lipschitz continuity of the boundary, which "when going beyond the scalar equations of second order" turns out to be a very natural class. These choices reflect the author's opinion that the Lame system and the biharmonic equations are just as important as the Laplace equation, and that the class of the domains with the Lipschitz continuous boundary (as opposed to smooth domains) is the most natural class of domains to consider in connection with these equations and their applications.

Partial Differential Equations

Jürgen Jost
Partial Differential Equations Book PDF

Author : Jürgen Jost
Publisher : Springer Science & Business Media
Page : 416 pages
File Size : 40,7 Mb
Release : 2012-11-13
Category : Mathematics
ISBN : 9781461448099

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Partial Differential Equations by Jürgen Jost Pdf

This book offers an ideal graduate-level introduction to the theory of partial differential equations. The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and develops estimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations. This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader. New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.

Partial Differential Equations with Variable Exponents

Vicentiu D. Radulescu,Dusan D. Repovs
Partial Differential Equations with Variable Exponents Book PDF

Author : Vicentiu D. Radulescu,Dusan D. Repovs
Publisher : CRC Press
Page : 323 pages
File Size : 48,6 Mb
Release : 2015-06-24
Category : Mathematics
ISBN : 9781498703444

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Partial Differential Equations with Variable Exponents by Vicentiu D. Radulescu,Dusan D. Repovs Pdf

Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type. The book presents the most important variational methods for elliptic PDEs described by nonhomogeneous differential operators and containing one or more power-type nonlinearities with a variable exponent. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear elliptic equations as well as their applications to various processes arising in the applied sciences. The analysis developed in the book is based on the notion of a generalized or weak solution. This approach leads not only to the fundamental results of existence and multiplicity of weak solutions but also to several qualitative properties, including spectral analysis, bifurcation, and asymptotic analysis. The book examines the equations from different points of view while using the calculus of variations as the unifying theme. Readers will see how all of these diverse topics are connected to other important parts of mathematics, including topology, differential geometry, mathematical physics, and potential theory.

Direct Methods in the Calculus of Variations

Enrico Giusti
Direct Methods in the Calculus of Variations Book PDF

Author : Enrico Giusti
Publisher : World Scientific
Page : 412 pages
File Size : 49,6 Mb
Release : 2003
Category : Mathematics
ISBN : 9789812795557

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Direct Methods in the Calculus of Variations by Enrico Giusti Pdf

This book provides a comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well known and have been widely used in the last century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this book, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The book is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory. Contents: Semi-Classical Theory; Measurable Functions; Sobolev Spaces; Convexity and Semicontinuity; Quasi-Convex Functionals; Quasi-Minima; HAlder Continuity; First Derivatives; Partial Regularity; Higher Derivatives. Readership: Graduate students, academics and researchers in the field of analysis and differential equations."