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Books
Applied Functional Analysis
- Author / Creator
- Available as
- Online
Details
- Creator
- Ammar Khanfer
- Format
- Books
- Language
- English
- Publication
-
- First edition
- Singapore : Springer Nature Singapore Pte Ltd, [2024]
- ©2024
- Physical Details
-
- 1 online resource (377 pages)
- ISBNs
- 9789819937882, 9789819937875
- Includes bibliographical references and index.
- Intro -- Preface -- Acknowledgments -- Contents -- About the Author -- 1 Operator Theory -- 1.1 Quick Review of Hilbert Space -- 1.1.1 Lebesgue Spaces -- 1.1.2 Convergence Theorems -- 1.1.3 Complete Space -- 1.1.4 Hilbert Space -- 1.1.5 Fundamental Mapping Theorems on Banach Spaces -- 1.2 The Adjoint of Operator -- 1.2.1 Bounded Linear Operators -- 1.2.2 Definition of Adjoint -- 1.2.3 Adjoint Operator on Hilbert Spaces -- 1.2.4 Self-adjoint Operators -- 1.3 Compact Operators -- 1.3.1 Definition and Properties of Compact Operators -- 1.3.2 The Integral Operator -- 1.3.3 Finite-Rank Operators -- 1.4 Hilbert-Schmidt Operator -- 1.4.1 Definition of Hilbert-Schmidt Operator -- 1.4.2 Basic Properties of HS Operators -- 1.4.3 Relations with Compact and Finite-Rank Operators -- 1.4.4 The Fredholm Operator -- 1.4.5 Characterization of HS Operators -- 1.5 Eigenvalues of Operators -- 1.5.1 Spectral Analysis -- 1.5.2 Definition of Eigenvalues and Eigenfunctions -- 1.5.3 Eigenvalues of Self-adjoint Operators -- 1.5.4 Eigenvalues of Compact Operators -- 1.6 Spectral Analysis of Operators -- 1.6.1 Resolvent and Regular Values -- 1.6.2 Bounded Below Mapping -- 1.6.3 Spectrum of Bounded Operator -- 1.6.4 Spectral Mapping Theorem -- 1.6.5 Spectrum of Compact Operators -- 1.7 Spectral Theory of Self-adjoint Compact Operators -- 1.7.1 Eigenvalues of Compact Self-adjoint Operators -- 1.7.2 Invariant Subspaces -- 1.7.3 Hilbert-Schmidt Theorem -- 1.7.4 Spectral Theorem For Self-adjoint Compact Operators -- 1.8 Fredholm Alternative -- 1.8.1 Resolvent of Compact Operators -- 1.8.2 Fundamental Principle -- 1.8.3 Fredholm Equations -- 1.8.4 Volterra Equations -- 1.9 Unbounded Operators -- 1.9.1 Introduction -- 1.9.2 Closed Operator -- 1.9.3 Basics Properties of Unbounded Operators -- 1.9.4 Toeplitz Theorem -- 1.9.5 Adjoint of Unbounded Operators
- 1.9.6 Deficiency Spaces of Unbounded Operators -- 1.9.7 Symmetry of Unbounded Operators -- 1.9.8 Spectral Properties of Unbounded Operators -- 1.10 Differential Operators -- 1.10.1 Green's Function and Dirac Delta -- 1.10.2 Laplacian Operator -- 1.10.3 Sturm-Liouville Operator -- 1.10.4 Momentum Operator -- 1.11 Problems -- 2 Distribution Theory -- 2.1 The Notion of Distribution -- 2.1.1 Motivation For Distributions -- 2.1.2 Test Functions -- 2.1.3 Definition of Distribution -- 2.2 Regular Distribution -- 2.2.1 Locally Integrable Functions -- 2.2.2 Notion of Regular Distribution -- 2.2.3 The Dual Space mathcalD -- 2.2.4 Basic Properties of Regular Distributions -- 2.3 Singular Distributions -- 2.3.1 Notion of Singular Distribution -- 2.3.2 Dirac Delta Distribution -- 2.3.3 Delta Sequence -- 2.3.4 Gaussian Delta Sequence -- 2.4 Differentiation of Distributions -- 2.4.1 Notion of Distributional Derivative -- 2.4.2 Calculus Rules -- 2.4.3 Examples of Distributional Derivatives -- 2.4.4 Properties of δ -- 2.5 The Fourier Transform Problem -- 2.5.1 Introduction -- 2.5.2 Fourier Transform on mathbbRn -- 2.5.3 Existence of Fourier Transform -- 2.5.4 Plancherel Theorem -- 2.6 Schwartz Space -- 2.6.1 Rapidly Decreasing Functions -- 2.6.2 Definition of Schwartz Space -- 2.6.3 Derivatives of Schwartz Functions -- 2.6.4 Isomorphism of Fourier Transform on Schwartz Spaces -- 2.7 Tempered Distributions -- 2.7.1 Definition of Tempered Distribution -- 2.7.2 Functions of Slow Growth -- 2.7.3 Examples of Tempered Distributions -- 2.8 Fourier Transform of Tempered Distribution -- 2.8.1 Motivation -- 2.8.2 Definition -- 2.8.3 Derivative of F.T. of Tempered Distribution -- 2.9 Inversion Formula of The Fourier Transform -- 2.9.1 Fourier Transform of Gaussian Function -- 2.9.2 Fourier Transform of Delta Distribution -- 2.9.3 Fourier Transform of Sign Function
- 2.10 Convolution of Distribution -- 2.10.1 Derivatives of Convolutions -- 2.10.2 Convolution in Schwartz Space -- 2.10.3 Definition of Convolution of Distributions -- 2.10.4 Fundamental Property of Convolutions -- 2.10.5 Fourier Transform of Convolution -- 2.11 Problems -- 3 Theory of Sobolev Spaces -- 3.1 Weak Derivative -- 3.1.1 Notion of Weak Derivative -- 3.1.2 Basic Properties of Weak Derivatives -- 3.1.3 Pointwise Versus Weak Derivatives -- 3.1.4 Weak Derivatives and Fourier Transform -- 3.2 Regularization and Smoothening -- 3.2.1 The Concept of Mollification -- 3.2.2 Mollifiers -- 3.2.3 Cut-Off Function -- 3.2.4 Partition of Unity -- 3.2.5 Fundamental Lemma of Calculus of Variations -- 3.3 Density of Schwartz Space -- 3.3.1 Convergence of Approximating Sequence -- 3.3.2 Approximations of mathcalS and Lp -- 3.3.3 Generalized Plancherel Theorem -- 3.4 Construction of Sobolev Spaces -- 3.4.1 Completion of Schwartz Spaces -- 3.4.2 Definition of Sobolev Space -- 3.4.3 Fractional Sobolev Space -- 3.5 Basic Properties of Sobolev Spaces -- 3.5.1 Convergence in Sobolev Spaces -- 3.5.2 Completeness and Reflexivity of Sobolev Spaces -- 3.5.3 Local Sobolev Spaces -- 3.5.4 Leibnitz Rule -- 3.5.5 Mollification with Sobolev Function -- 3.5.6 W0k,p(Ω) -- 3.6 W1,p(Ω) -- 3.6.1 Absolute Continuity Characterization -- 3.6.2 Inclusions -- 3.6.3 Chain Rule -- 3.6.4 Dual Space of W1,p(Ω) -- 3.7 Approximation of Sobolev Spaces -- 3.7.1 Local Approximation -- 3.7.2 Global Approximation -- 3.7.3 Consequences of Meyers-Serrin Theorem -- 3.8 Extensions -- 3.8.1 Motivation -- 3.8.2 The Zero Extension -- 3.8.3 Coordinate Transformations -- 3.8.4 Extension Operator -- 3.9 Sobolev Inequalities -- 3.9.1 Sobolev Exponent -- 3.9.2 Fundamental Inequalities -- 3.9.3 Gagliardo-Nirenberg-Sobolev Inequality -- 3.9.4 Poincare Inequality -- 3.9.5 Estimate for W1,p
- 3.9.6 The Case p=n -- 3.9.7 Holder Spaces -- 3.9.8 The Case p> -- n -- 3.9.9 General Sobolev Inequalities -- 3.10 Embedding Theorems -- 3.10.1 Compact Embedding -- 3.10.2 Rellich-Kondrachov Theorem -- 3.10.3 High Order Sobolev Estimates -- 3.10.4 Sobolev Embedding Theorem -- 3.10.5 Embedding of Fractional Sobolev Spaces -- 3.11 Problems -- 4 Elliptic Theory -- 4.1 Elliptic Partial Differential Equations -- 4.1.1 Elliptic Operator -- 4.1.2 Uniformly Elliptic Operator -- 4.1.3 Elliptic PDEs -- 4.2 Weak Solution -- 4.2.1 Motivation for Weak Solutions -- 4.2.2 Weak Formulation of Elliptic BVP -- 4.2.3 Classical Versus Strong Versus Weak Solutions -- 4.3 Poincare Equivalent Norm -- 4.3.1 Poincare Inequality on H01 -- 4.3.2 Equivalent Norm on H01 -- 4.3.3 Poincare-Wirtinger Inequality -- 4.3.4 Quotient Sobolev Space -- 4.4 Elliptic Estimates -- 4.4.1 Bilinear Forms -- 4.4.2 Elliptic Bilinear Mapping -- 4.4.3 Garding's Inequality -- 4.5 Symmetric Elliptic Operators -- 4.5.1 Riesz Representation Theorem for Hilbert Spaces -- 4.5.2 Existence and Uniqueness Theorem-Poisson's Equation -- 4.5.3 Existence and Uniqueness Theorem-Helmholtz Equation -- 4.5.4 Ellipticity and Coercivity -- 4.5.5 Existence and Uniqueness Theorem-Symmetric Uniformly Operator -- 4.6 General Elliptic Operators -- 4.6.1 Lax-Milgram Theorem -- 4.6.2 Dirichlet Problems -- 4.6.3 Neumann Problems -- 4.7 Spectral Properties of Elliptic Operators -- 4.7.1 Resolvent of Elliptic Operators -- 4.7.2 Fredholm Alternative for Elliptic Operators -- 4.7.3 Spectral Theorem for Elliptic Operators -- 4.8 Self-adjoint Elliptic Operators -- 4.8.1 The Adjoint of Elliptic Bilinear -- 4.8.2 Eigenvalue Problem of Elliptic Operators -- 4.8.3 Spectral Theorem of Elliptic Operator -- 4.9 Regularity for the Poisson Equation -- 4.9.1 Weyl's Lemma -- 4.9.2 Difference Quotients -- 4.9.3 Caccioppoli's Inequality
- 4.9.4 Interior Regularity for Poisson Equation -- 4.10 Regularity for General Elliptic Equations -- 4.10.1 Interior Regularity -- 4.10.2 Higher Order Interior Regularity -- 4.10.3 Interior Smoothness -- 4.10.4 Boundary Regularity -- 4.11 Problems -- 5 Calculus of Variations -- 5.1 Minimization Problem -- 5.1.1 Definition of Minimization Problem -- 5.1.2 Lower Semicontinuity -- 5.1.3 Minimization Problems in Finite-Dimensional Spaces -- 5.1.4 Convexity -- 5.1.5 Minimization in Infinite-Dimensional Space -- 5.2 Weak Topology -- 5.2.1 Notion of Weak Topology -- 5.2.2 Weak Convergence -- 5.2.3 Weakly Closed Sets -- 5.2.4 Reflexive Spaces -- 5.2.5 Weakly Lower Semicontinuity -- 5.3 Direct Method -- 5.3.1 Direct Verses Indirect Methods -- 5.3.2 Minimizing Sequence -- 5.3.3 Procedure of Direct Method -- 5.3.4 Coercivity -- 5.3.5 The Main Theorem on the Existence of Minimizers -- 5.4 The Dirichlet Problem -- 5.4.1 Variational Integral -- 5.4.2 Dirichlet Principle -- 5.4.3 Weierstrass Counterexample -- 5.5 Dirichlet Principle in Sobolev Spaces -- 5.5.1 Minimizer of the Dirichlet Integral in H01 -- 5.5.2 Minimizer of the Dirichlet Integral in H1 -- 5.5.3 Dirichlet Principle -- 5.5.4 Dirichlet Principle with Neumann Condition -- 5.5.5 Dirichlet Principle with Neumann B.C. in Sobolev Spaces -- 5.6 Gateaux Derivatives of Functionals -- 5.6.1 Introduction -- 5.6.2 Historical Remark -- 5.6.3 Gateaux Derivative -- 5.6.4 Basic Properties of G-Derivative -- 5.6.5 G-Differentiability and Continuity -- 5.6.6 Frechet Derivative -- 5.6.7 G-Differentiability and Convexity -- 5.6.8 Higher Gateaux Derivative -- 5.6.9 Minimality Condition -- 5.7 Poisson Variational Integral -- 5.7.1 Gateaux Derivative of Poisson Integral -- 5.7.2 Symmetric Elliptic PDEs -- 5.7.3 Dirichlet Principle of Symmetric Elliptic PDEs -- 5.8 Euler-Lagrange Equation -- 5.8.1 Lagrangian Integral
- 5.8.2 First Variation