Description 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 Co-semigroups, as the Hille-Yosida 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 finite-dimensional-like methods
for certain semilinear pseudo-parabolic, 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 Klein-Gordon 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 self-contained, requiring
only a basic course in Functional Analysis and Partial Differential Equations.
Audience
Institutes and Departments of Mathematics. Departments of Physics, Libraries of Universities.
Contents 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.
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)=ξ.
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 (λI-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)=ξ.
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.
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.
Books and book related electronic products are priced in US dollars (USD), euro (EUR), and Great Britain Pounds (GBP). USD prices apply to the Americas and Asia Pacific. EUR prices apply in Europe and the Middle East. GBP prices apply to the UK and all other countries.
Home | Elsevier Sites | Privacy Policy | Terms and Conditions | Feedback | Site Map | A Reed Elsevier Company