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Co-Semigroups and Applications als Buch

Co-Semigroups and Applications

Sprache: Englisch.
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Contains a unitary and systematic presentation of parts of a fundamental branch of functional analysis: linear semigroup theory with main emphasis on examples and applications. This book includes several results referring to various classes of semigr... weiterlesen


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Co-Semigroups and Applications als Buch
Titel: Co-Semigroups and Applications
Autor/en: Ioan I. Vrabie

ISBN: 0444512888
EAN: 9780444512888
Sprache: Englisch.

April 2003 - gebunden - 396 Seiten


Contains a unitary and systematic presentation of parts of a fundamental branch of functional analysis: linear semigroup theory with main emphasis on examples and applications. This book includes several results referring to various classes of semigroups such as equicontinuous, compact, differentiable and analytic.


Preface. Chapter 1. Preliminaries. 1.1 Vector-Valued Measurable Functions. 1.2 The Bochner Integral. 1.3 Basic Function Spaces. 1.4 Functions of Bounded Variation. 1.5 Sobolev Spaces. 1.6 Unbounded Linear Operators. 1.7 Elements of Spectral Analysis. 1.8 Functional Calculus for Bounded Operators. 1.9 Functional Calculus for Unbounded Operators. Problems. Notes. Chapter 2. Semigroups of Linear Operators 2.1 Uniformly Continuous Semigroups. 2.2 Generators of Uniformly Continuous Semigroups. 2.3 Co-Semigroups. General Properties. 2.4 The Infinitesimal Generator. Problems. Notes. Chapter 3. Generation Theorems 3.1 The Hille-Yosida Theorem. Necessity. 3.2 The Hille-Yosida Theorem. Sufficiency. 3.3 The Feller-Miyadera-Phillips Theorem. 3.4 The Lumer-Phillips Theorem. 3.5 Some Consequences. 3.6 Examples. 3.7 The Dual of a Co-Semigroup. 3.8 The Sun Dual of a Co-Semigroup. 3.9 Stone Theorem. Problems. Notes. Chapter 4. Differential Operators Generating Co-Semigroups 4.1 The Laplace Operator with Dirichlet Boundary Conditions. 4.2 The Laplace Operator with Neumann Boundary Condition. 4.3 The Maxwell Operator. 4.4 The Directional Derivative. 4.5 The Schroedinger Operator. 4.6 The Wave Operator. 4.7 The Airy Operator. 4.8 The Equations of Linear Thermoelasticity. 4.9 The Equations of Linear Viscoelasticity. Problems. Notes. Chapter 5. Approximation Problems and Applications 5.1 The Continuity of A etA. 5.2 The Chernoff and Lie-Trotter Formulae. 5.3 A Perturbation Result. 5.4 The Central Limit Theorem. 5.5 Feynman Formula. 5.6 The Mean Ergodic Theorem. Problems. Notes. Chapter 6. Some Special Classes of Co-Semigroups 6.1 Equicontinuous Semigroups. 6.2 Compact Semigroups. 6.3 Differentiable Semigroups. 6.4 Semigroups with Symmetric Generators. 6.5 The Linear Delay Equation. Problems. Notes. Chapter 7. Analytic Semigroups. 7.1 Definition and Characterizations. 7.2 The Heat Equation. 7.3 The Stokes Equation. 7.4 A Parabolic Problem with Dynamic Boundary Conditions. 7.5 An Elliptic Problem with Dynamic Boundary Conditions. 7.6 Fractional Powers of Closed Operators. 7.7 Further Investigations in the Analytic Case. Problems. Notes. Chapter 8. The Nonhomogeneous Cauchy Problem 8.1 The Cauchy Problem u'=Au+f, u(a)= . 8.2 Smoothing Effect. The Hilbert Space Case. 8.3 Compactness of the Solution Operator from Lp(a,b;X). 8.4 The Case when ( I-A) -1 is Compact. 8.5 Compactness of the Solution Operator from Ll(a,b;X). Problems. Notes. Chapter 9. Linear Evolution Problems with Measures as Data 9.1 The Problem du={Au}dt+dg, u(a)= . 9.2 Regularity of L -Solutions. 9.3 A Characterization of L -Solutions. 9.4 Compactness of the L -Solution Operator. 9.5 Evolution Equations with "Spatial" Measures as Data. Problems. Notes. Chapter 10. Some Nonlinear Cauchy Problems 10.1 Peano's Local Existence Theorem. 10.2 The Problem u'=f(t,u)+g(t,u). 10.3 Saturated Solutions. 10.4 The Klein-Gordon Equation. 10.5 An Application to a Problem in Mechanics. Problems. Notes. Chapter 11. The Cauchy Problem for Semilinear Equations 11.1 The Problem u'=Au+f(t,u) with f Lipschitz. 11.2 The Problem u'=Au+f(t,u) with f Continuous. 11.3 Saturated Solutions. 11.4 Asymptotic Behaviour. 11.5 The Klein-Gordon Equation Revisited. 11.6 A Parabolic Semilinear Equation. Problems. Notes. Chapter 12. Semilinear Equations Involving Measures 12.1 The Problem du={Au}dt+dgu with u gu Lipschitz. 12.2 The Problem du={Au}dt+dgu with u gu Continuous. 12.3 Saturated L -Solutions. 12.4 The Case of Spatial Measures. 12.5 Two Examples. 12.6 One More Example. Problems. Notes. Appendix A. Compactness Results A.1 Compact operators. A.2 Compactness in C([a,b]; X). A.3 Compactness in C([a,b]; Xw). A.4 Compactness in LP(a,b; X). A.5 Compactness in LP(a,b; X). Continued. A.6 The Superposition Operator. Problems. Notes. Solutions. Bibliography. List of Symbols. Subject Index.


"The book is self-contained, requires only some acquaintance with functional analysis and partial differential equations." Hana Petzeltova. Mathematica Bohemica, 2003.

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