Linear Sobolev Type Equations and Degenerate Semigroups of Operators

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ISBN: 9789067643832 | 9067643831
Cover: Hardcover
Copyright: 4/1/2003

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Focusing on the mathematics, and providing only a minimum of explicatory comment, this volume contains six chapters covering auxiliary material, relatively p-radial operators, relatively p-sectorial operators, relatively ?-bounded operators, Cauchy problems for inhomogenous Sobolev-type equations, bounded solutions to Sobolev-type equations, and optimal control.

Auxiliary material

(12)

Banach spaces and linear operators

(5)

Theorems on infinitesimal generators

(3)

Functional spaces and differential operators

(4)

Relatively p-radial operators and degenerate strongly continuous semigroups of operators

(42)

Introduction

(2)

Relative resolvents

(5)

Relatively p-radial operators

(5)

Degenerate strongly continuous semigroups of operators

(6)

Approximations of Hille---Widder---Post type

(5)

Splitting of spaces

(6)

Infinitesimal generators and phase spaces

(4)

Generators of degenerate strongly continuous semigroups of operators

(3)

Degenerate strongly continuous groups of operators

(6)

Relatively p-sectorial operators and degenerate analytic semigroups of operators

(32)

Introduction

(2)

Relatively p-sectorial operators

(2)

Degenerate analytic semigroups of operators

(6)

Phase spaces for the case of degenerate analytic semigroups

(3)

Space splitting

(6)

Generators of degenerate analytic semigroups of operators

(3)

Degenerate infinitely differentiable semigroups of operators

(4)

Phase spaces for the case of degenerate infinitely continuously differentiable semigroups

(3)

Kernels and images of degenerate infinitely differentiable semigroups of operators

(3)

Relatively σ-bounded operators and degenerate analytic groups of operators

(32)

Introduction

(2)

Relatively σ-bounded operators

(3)

Relative σ-boundedness and relative p-sectoriality

(3)

Relative σ-boundedness and relatively adjoint vectors

(2)

Degenerate analytical groups of operators

(4)

Sufficient conditions of the relative σ-bounded ness

101

(7)

The case of a Fredholm operator

108

(3)

Analytical semigroups of operators degenerating on the chains of relatively adjoint vectors of an arbitrary length

111

(8)

Cauchy problem for inhomogeneous Sobolev-type equations

119

(40)

Introduction

119

(2)

Case of a relatively σ-bounded operator

121

(10)

The case of a relatively p-sectorial operator

131

(4)

Case of a relatively p-radial operator

135

(4)

Strong solution of Cauchy problem

139

(4)

Cauchy problem for an equation with Banach-adjoint operators

143

(7)

Propagators

150

(4)

Inhomogeneous Cauchy problem for high-order Sobolev-type equations

154

(5)

Bounded solutions of Sobolev-type equations

159

(24)

Introduction

159

(1)

Relatively spectral theorem

160

(3)

Bounded relaxed solutions of a homogeneous equation

163

(7)

Bounded solutions of the inhomogeneous equation

170

(7)

Examples

177

(6)

Optimal control

183

(18)

Introduction

183

(1)

Strong solution of Cauchy problem for an equation with Hilbert-adjoint operators

183

(5)

Problem of optimal control for an equation with relatively σ-bounded operator

188

(4)

Problem of optimal control for equation with a relatively p-sectorial operator

192

(2)

Barenblatt---Zheltov---Kochina equation

194

(1)

System of ordinary differential equations

195

(2)

Equation of the evolution of the free filtered-fluid surface

197

(4)

Bibliography

201

(14)

Index

215

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Linear Sobolev Type Equations and Degenerate Semigroups of Operators

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Table of Contents

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