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Elliptic Equations: An Introductory Course / by Michel Chipot; edited by Herbert Amann, Steven G. Krantz, Shrawan Kumar, Jan Nekovář
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Birkhäuser Advanced Texts / Basler Lehrbücher | SpringerLink BücherPublisher: Basel ; Boston, Mass. ; Berlin : Birkhäuser, 2009Publisher: [Heidelberg] : [Springer], 2009Description: Online-Ressource (digital)ISBN:- 9783764399825
- 515.353
- 515 23
- 515.3533
- 510
- QA370-380
- QA377
Contents:
Summary: Basic Techniques -- Hilbert Space Techniques -- A Survey of Essential Analysis -- Weak Formulation of Elliptic Problems -- Elliptic Problems in Divergence Form -- Singular Perturbation Problems -- Asymptotic Analysis for Problems in Large Cylinders -- Periodic Problems -- Homogenization -- Eigenvalues -- Numerical Computations -- More Advanced Theory -- Nonlinear Problems -- L?-estimates -- Linear Elliptic Systems -- The Stationary Navier—Stokes System -- Some More Spaces -- Regularity Theory -- The p-Laplace Equation -- The Strong Maximum Principle -- Problems in the Whole Space.Summary: The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and refinements. Apart from the basic theory of equations in divergence form it includes subjects such as singular perturbation problems, homogenization, computations, asymptotic behaviour of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes system, p-Laplace equation. Just a minimum on Sobolev spaces has been introduced, and work or integration on the boundary has been carefully avoided to keep the reader's attention on the beauty and variety of these issues. The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original and have not been published elsewhere. The book will be of interest to graduate students and faculty members specializing in partial differential equations.PPN: PPN: 1647816327Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Contents""; ""Preface""; ""Part I Basic Techniques""; ""Chapter 1 Hilbert Space Techniques""; ""1.1 The projection on a closed convex set""; ""1.2 The Riesz representation theorem""; ""1.3 The Lax�Milgram theorem""; ""1.4 Convergence techniques""; ""Exercises""; ""Chapter 2 A Survey of Essential Analysis""; ""2.1 Lp-techniques""; ""2.2 Introduction to distributions""; ""2.3 Sobolev Spaces""; ""Exercises""; ""Chapter 3 Weak Formulation of Elliptic Problems""; ""3.1 Motivation""; ""3.2 The weak formulation""; ""Exercises""; ""Chapter 4 Elliptic Problems in Divergence Form""
""4.1 Weak formulation""""4.2 The weak maximum principle""; ""4.3 Inhomogeneous problems""; ""Exercises""; ""Chapter 5 Singular Perturbation Problems""; ""5.1 A prototype of a singular perturbation problem""; ""5.2 Anisotropic singular perturbation problems""; ""Exercises""; ""Chapter 6 Asymptotic Analysis for Problems in Large Cylinders""; ""6.1 A model problem""; ""6.2 Another type of convergence""; ""6.3 The general case""; ""6.4 An application""; ""Exercises""; ""Chapter 7 Periodic Problems""; ""7.1 A general theory""; ""7.2 Some additional remarks""; ""Exercises""
""Chapter 8 Homogenization""""8.1 More on periodic functions""; ""8.2 Homogenization of elliptic equations""; ""Exercises""; ""Chapter 9 Eigenvalues""; ""9.1 The one-dimensional case""; ""9.2 The higher-dimensional case""; ""9.3 An application""; ""Exercises""; ""Chapter 10 Numerical Computations""; ""10.1 The finite difference method""; ""10.2 The finite element method""; ""Exercises""; ""Part II More Advanced Theory""; ""Chapter 11 Nonlinear Problems""; ""11.1 Monotone methods""; ""11.2 Quasilinear equations""; ""11.3 Nonlocal problems""; ""11.4 Variational inequalities""; ""Exercises""
""Chapter 12 L-estimates""""12.1 Some simple cases""; ""12.2 A more involved estimate""; ""12.3 The Sobolev�Gagliardo�Nirenberg inequality""; ""12.4 The maximum principle on small domains""; ""Exercises""; ""Chapter 13 Linear Elliptic Systems""; ""13.1 The general framework""; ""13.2 Some examples""; ""Exercises""; ""Chapter 14 The Stationary Navier�Stokes System""; ""14.1 Introduction""; ""14.2 Existence and uniqueness result""; ""Exercise""; ""Chapter 15 Some More Spaces""; ""15.1 Motivation""; ""15.2 Essential features of the Sobolev spaces""; ""15.3 An application""; ""Exercises""
""Chapter 16 Regularity Theory""""16.1 Introduction""; ""16.2 The translation method""; ""16.3 Regularity of functions in Sobolev spaces""; ""16.4 The bootstrap technique""; ""Exercises""; ""Chapter 17 The p-Laplace Equation""; ""17.1 A minimization technique""; ""17.2 A weak maximum principle and its consequences""; ""17.3 A generalization of the Lax�Milgram theorem""; ""Exercises""; ""Chapter 18 The Strong Maximum Principle""; ""18.1 A first version of the maximum principle""; ""18.2 The Hopf maximum principle""; ""18.3 Application: the moving plane technique""; ""Exercises""
""Chapter 19 Problems in the Whole Space""
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