Format:
Online-Ressource
ISBN:
9783034800877
Series Statement:
Monographs in Mathematics 96
Content:
This monograph gives a systematic account of the theory of vector-valued Laplace transforms, ranging from representation theory to Tauberian theorems. In parallel, the theory of linear Cauchy problems and semigroups of operators is developed completely in the spirit of Laplace transforms. Existence and uniqueness, regularity, approximation and above all asymptotic behaviour of solutions are studied. Diverse applications to partial differential equations are given. The book contains an introduction to the Bochner integral and several appendices on background material. It is addressed to student
Note:
Description based upon print version of record
,
Vector-valued Laplace Transforms and CauchyProblems; Contents; Prefaces; Preface to the First Edition; Preface to the Second Edition; Part I Laplace Transforms and Well-Posedness of Cauchy Problems; Chapter 1 The Laplace Integral; 1.1 The Bochner Integral; 1.2 The Radon-Nikodym Property; 1.3 Convolutions; 1.4 Existence of the Laplace Integral; 1.5 Analytic Behaviour; 1.6 Operational Properties; 1.7 Uniqueness, Approximation and Inversion; 1.8 The Fourier Transform and Plancherel's Theorem; 1.9 The Riemann-Stieltjes Integral; 1.10 Laplace-Stieltjes Integrals; 1.11 Notes
,
Chapter 2 The Laplace Transform2.1 Riesz-Stieltjes Representation; 2.2 A Real Representation Theorem; 2.3 Real and Complex Inversion; 2.4 Transforms of Exponentially Bounded Functions; 2.5 Complex Conditions; 2.6 Laplace Transforms of Holomorphic Functions; 2.7 Completely Monotonic Functions; 2.8 Notes; Chapter 3 Cauchy Problems; 3.1 C0-semigroups and Cauchy Problems; 3.2 Integrated Semigroups and Cauchy Problems; 3.3 Real Characterization; 3.4 Dissipative Operators; 3.5 Hille-Yosida Operators; 3.6 Approximation of Semigroups; 3.7 Holomorphic Semigroups; 3.8 Fractional Powers
,
3.9 Boundary Values of Holomorphic Semigroups3.10 Intermediate Spaces; 3.11 Resolvent Positive Operators; 3.12 Complex Inversion and UMD-spaces; 3.13 Norm-continuous Semigroups and Hilbert Spaces; 3.14 The Second Order Cauchy Problem; 3.15 Sine Functions and Real Characterization; 3.17 Notes; Part II Tauberian Theorems and Cauchy Problems; Chapter 4 Asymptotics of Laplace Transforms; 4.1 Abelian Theorems; 4.2 Real Tauberian Theorems; 4.3 Ergodic Semigroups; 4.4 Complex Tauberian Theorems: the Contour Method; 4.5 Almost Periodic Functions; 4.6 Countable Spectrum and Almost Periodicity
,
4.7 Asymptotically Almost Periodic Functions4.8 Carleman Spectrum and Fourier Transform; 4.9 Complex Tauberian Theorems: the Fourier Method; 4.10 Notes; Chapter 5 Asymptotics of Solutions of Cauchy Problems; 5.1 Growth Bounds and Spectral Bounds; 5.2 Semigroups on Hilbert Spaces; 5.3 Positive Semigroups; 5.4 Splitting Theorems; 5.5 Countable Spectral Conditions; 5.6 Solutions of Inhomogeneous Cauchy Problems; 5.7 Notes; Part III Applications and Examples; Chapter 6 The Heat Equation; 6.1 The Laplacian with Dirichlet Boundary Conditions; 6.2 Inhomogeneous Boundary Conditions
,
6.3 Asymptotic Behaviour6.4 Notes; Chapter 7 The Wave Equation; 7.1 Perturbation of Selfadjoint Operators; 7.2 The Wave Equation in L²(Ω); 7.3 Notes; Chapter 8 Translation Invariant Operatorson Lp (Rn); 8.1 Translation Invariant Operators and C0-semigroups; 8.2 Fourier Multipliers; 8.3 Lp-spectra and Integrated Semigroups; 8.4 Systems of Differential Operators on Lp-spaces; 8.5 Notes; Appendix A Vector-valued Holomorphic Functions; Appendix B Closed Operators; Appendix C Ordered Banach Spaces; Appendix D Banach Spaces which Contain c0; Appendix E Distributions and Fourier Multipliers
,
Bibliography
Language:
English
Subjects:
Mathematics
Keywords:
Evolutionsgleichung
;
Laplace-Transformation
;
Cauchy-Anfangswertproblem
;
Electronic books
Author information:
Arendt, Wolfgang 1950-
Bookmarklink
