Co Semigroups and Applications
By Ioan I. Vrabie
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 emphasis on examples and applications. There are several specialized, but quite interesting, topics which didn't find their place into a monograph till now, mainly because they are very new. So, the book, although containing the main parts of the classical theory of Cosemigroups, as the HilleYosida theory, includes also several very new results, as for instance those referring to various classes of semigroups such as equicontinuous, compact, differentiable, or analytic, as well as to some nonstandard types of partial differential equations, i.e. elliptic and parabolic systems with dynamic boundary conditions, and linear or semilinear differential equations with distributed (time, spatial) measures. Moreover, some finitedimensionallike methods for certain semilinear pseudoparabolic, or hyperbolic equations are also disscussed. Among the most interesting applications covered are not only the standard ones concerning the Laplace equation subject to either Dirichlet, or Neumann boundary conditions, or the Wave, or KleinGordon equations, but also those referring to the Maxwell equations, the equations of Linear Thermoelasticity, the equations of Linear Viscoelasticity, to list only a few. Moreover, each chapter contains a set of various problems, all of them completely solved and explained in a special section at the end of the book.
The 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 selfcontained, requiring only a basic course in Functional Analysis and Partial Differential Equations.
Audience
Institutes and Departments of Mathematics. Departments of Physics, Libraries of Universities.
NorthHolland Mathematics Studies
Hardbound, 396 Pages
Published: March 2003
Imprint: Jai Press (elsevier)
ISBN: 9780444512888
Reviews

"The book is selfcontained, requires only some acquaintance with functional analysis and partial differential equations."
Hana Petzeltova. Mathematica Bohemica, 2003.
Contents
Preface.
Chapter 1. Preliminaries.
1.1 VectorValued 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 CoSemigroups. General Properties.
2.4 The Infinitesimal Generator.
Problems.
Notes.Chapter 3. Generation Theorems
3.1 The HilleYosida Theorem. Necessity.
3.2 The HilleYosida Theorem. Sufficiency.
3.3 The FellerMiyaderaPhillips Theorem.
3.4 The LumerPhillips Theorem.
3.5 Some Consequences.
3.6 Examples.
3.7 The Dual of a CoSemigroup.
3.8 The Sun Dual of a CoSemigroup.
3.9 Stone Theorem.
Problems.
Notes.Chapter 4. Differential Operators Generating CoSemigroups
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 LieTrotter 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 CoSemigroups
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 Lp(a,b;X).
8.4 The Case when (&lgr;IA) 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)=&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 KleinGordon 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 KleinGordon 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.