Linear Sobolev type equations and degenerate semigroups of operators

Linear Sobolev type equations and degenerate semigroups of operators / G.A. Sviridyuk and V.E. Fedorov.

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Bibliographic Details
Main Author: Sviridyuk, G. A.
Other Authors: Fedorov, V. E.
Format: eBook
Language:English
Published: Utrecht ; Boston : VSP, 2003.
Series:Inverse and ill-posed problems series.
Subjects:
Sobolev spaces.
Differential equations, Linear.
Degenerate differential equations.
Semigroups of operators.
MATHEMATICS > Calculus.
MATHEMATICS > Mathematical Analysis.
Degenerate differential equations
Differential equations, Linear
Semigroups of operators
Sobolev spaces
Online Access:Click for online access
Table of Contents:
  • Chapter 1. Auxiliary material 1
  • 1.1. Banach spaces and linear operators 1
  • 1.2. Theorems on infinitesimal generators 6
  • 1.3. Functional spaces and differential operators 9
  • Chapter 2. Relatively p-radial operators and degenerate strongly continuous semigroups of operators 13
  • 2.1. Relative resolvents 15
  • 2.2. Relatively p-radial operators 20
  • 2.3. Degenerate strongly continuous semigroups of operators 25
  • 2.4. Approximations of Hille
  • Widder
  • Post type 31
  • 2.5. Splitting of spaces 36
  • 2.6. Infinitesimal generators and phase spaces 42
  • 2.7. Generators of degenerate strongly continuous semigroups of operators 46
  • 2.8. Degenerate strongly continuous groups of operators 49
  • Chapter 3. Relatively p-sectorial operators and degenerate analytic semigroups of operators 55
  • 3.1. Relatively p-sectorial operators 57
  • 3.2. Degenerate analytic semigroups of operators 59
  • 3.3. Phase spaces for the case of degenerate analytic semigroups 65
  • 3.4. Space splitting 68
  • 3.5. Generators of degenerate analytic semigroups of operators 74
  • 3.6. Degenerate infinitely differentiable semigroups of operators 77
  • 3.7. Phase spaces for the case of degenerate infinitely continuously differentiable semigroups 81
  • 3.8. Kernels and images of degenerate infinitely differentiable semigroups of operators 84
  • Chapter 4. Relatively [sigma]-bounded operators and degenerate analytic groups of operators 87
  • 4.1. Relatively [sigma]-bounded operators 89
  • 4.2. Relative [sigma]-boundedness and relative p-sectoriality 92
  • 4.3. Relative [sigma]-boundedness and relatively adjoint vectors 95
  • 4.4. Degenerate analytical groups of operators 97
  • 4.5. Sufficient conditions of the relative [sigma]-bounded ness 101
  • 4.6. The case of a Fredholm operator 108
  • 4.7. Analytical semigroups of operators degenerating on the chains of relatively adjoint vectors of an arbitrary length 111
  • Chapter 5. Cauchy problem for inhomogeneous Sobolev-type equations 119
  • 5.1. Case of a relatively [sigma]-bounded operator 121
  • 5.2. The case of a relatively p-sectorial operator 131
  • 5.3. Case of a relatively p-radial operator 135
  • 5.4. Strong solution of Cauchy problem 139
  • 5.5. Cauchy problem for an equation with Banach-adjoint operators 143
  • 5.6. Propagators 150
  • 5.7. Inhomogeneous Cauchy problem for high-order Sobolev-type equations 154
  • Chapter 6. Bounded solutions of Sobolev-type equations 159
  • 6.1. Relatively spectral theorem 160
  • 6.2. Bounded relaxed solutions of a homogeneous equation 163
  • 6.3. Bounded solutions of the inhomogeneous equation 170
  • Chapter 7. Optimal control 183
  • 7.1. Strong solution of Cauchy problem for an equation with Hilbert-adjoint operators 183
  • 7.2. Problem of optimal control for an equation with relatively [sigma]-bounded operator 188
  • 7.3. Problem of optimal control for equation with a relatively p-sectorial operator 192
  • 7.4. Barenblatt
  • Zheltov
  • Kochina equation 194
  • 7.5. System of ordinary differential equations 195
  • 7.6. Equation of the evolution of the free filtered-fluid surface 197.

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