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# C<INF>o</INF>-Semigroups and Applications

- 1st Edition, Volume 191 - March 21, 2003
- Author: Ioan I. Vrabie
- Language: English
- Hardback ISBN:9 7 8 - 0 - 4 4 4 - 5 1 2 8 8 - 8
- eBook ISBN:9 7 8 - 0 - 0 8 - 0 5 3 0 0 4 - 8

The book contains a unitary and systematic presentation of both classical and very recent parts of a fundamental branch of functional analysis: linear semigroup theory with main… Read more

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Request a sales quoteThe book is primarily addressed to graduate students and researchers in the field, but it would be of interest for both physicists and engineers. It should be emphasised that it is almost self-contained, requiring only a basic course in Functional Analysis and Partial Differential Equations.

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* → *e ^{tA}*. 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)=&xgr;*. 8.2 Smoothing Effect. The Hilbert Space Case.8.3 Compactness of the Solution Operator from *L ^{p}(a,b;X)*.8.4 The Case when

*(&lgr;I-A)*

^{-1}is Compact.8.5 Compactness of the Solution Operator from

*L*.Problems.Notes.

^{l}(a,b;X)Chapter 9. Linear Evolution Problems with Measures as Data

9.1 The Problem *du={Au}dt+dg, u(a)=&xgr;*. 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 *L ^{P}(a,b; X)*.A.5 Compactness in

*L*. Continued.A.6 The Superposition Operator.Problems.Notes.

^{P}(a,b; X)Solutions.Bibliography. List of Symbols.Subject Index.

- No. of pages: 396
- Language: English
- Edition: 1
- Volume: 191
- Published: March 21, 2003
- Imprint: JAI Press
- Hardback ISBN: 9780444512888
- eBook ISBN: 9780080530048

IV

### Ioan I. Vrabie

*C<INF>o</INF>-Semigroups and Applications*on ScienceDirect