Bibliothèque de L'IMIST/CNRST

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 C<INF>o</INF>-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, id est 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.

Includes bibliographical references (p. 361-367) and index.

Cover -- 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. C0-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 C0-Semigroup -- 3.8. The Sun Dual of a C0-Semigroup -- 3.9. Stone Theorem -- Problems -- Notes -- Chapter 4. Differential Operators Generating C0- Semigroups -- 4.1. The Laplace Operator with Dirichlet Boundary Condition -- 4.2. The Laplace Operator with Neumann Boundary Condition -- 4.3. The Maxwell Operator -- 4.4. The Directional Derivative -- 4.5. The Schrödinger 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 C0-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. An Approximation Result -- 8.4. Compactness of the Solution Operator from LP(a, b ; X -- 8.5. The Case when (?I -- A) -1 is Compact -- 8.6. Compactness of the Sol.