线性发展方程的单参数半群

书籍目录

preface

prelude

i. linear dynamical systems

1. cauchy's functional equation

2. finite-dimensional systems: matrix semigroups

3. uniformly continuous operator semigroups

4. more semigroups

a. multiplication semigroups on co(fi)

b. multiplication semigroups on lp(ω,μ)

c. translation semigroups

5. strongly continuous semigroups

a. basic properties

b. standard constructions

notes

ii. semigroups, generators, and resolvents

1. generators of semigroups and their resolvents

2. examples revisited

a. standard constructions

b. standard examples

3. hille-yosida generation theorems

a. generation of groups and semigroups

b. dissipative operators and contraction semigroups

c. more examples

4. special classes of semigroups

a. analytic semigroups

b. differentiable semigroups

c. eventually norm-continuons semigroups

d. eventually compact semigroups

e. examples

5. interpolation and extrapolation spaces for semigroups

simon brendle

a. sobolev towers

b. favard and abstract h61der spaces

c. fractional powers

6. well-posedness for evolution equations

notes

iii. perturbation and approximation of semigroups

1. bounded perturbations

2. perturbations of contractive and analytic semigroups

3. more perturbations

a. the perturbation theorem of desch-schappacher

b. comparison of semigroups

c. the perturbation theorem of miyadera-voigt

d. additive versus multiplicative perturbations

4. trotter-kato approximation theorems

a. a technical tool: pseudoresolvents

b. the approximation theorems

c. examples

5. approximation formulas

a. chernoff product formula

b. inversion formulas

notes

iv. spectral theory for semigroups and generators

1. spectral theory for closed operators

2. spectrum of semigroups and generators

a. basic theory

b. spectrum of induced semigroups

c. spectrum of periodic semigroups

3. spectral mapping theorems

a. examples and counterexamples

b. spectral mapping theorems for semigroups

c. weak spectral mapping theorem for bounded groups

4. spectral theory and perturbation

notes

v. asymptotics of semigroups

1. stability and hyperbolicity for semigroups

a. stability concepts

b. characterization of uniform exponential stability

c. hyperbolic decompositions

2. compact semigroups

a. general semigroups

b. weakly compact semigroups

c. strongly compact semigroups

3. eventually compact and quasi-compact semigroups

4. mean ergodic semigroups

notes

vi. semigroups everywhere

1. semigroups for population equations

a. semigroup method for the cell equation

b. intermezzo on positive semigroups

c. asymptotics for the cell equation

notes

2. semigroups for the transport equation

a. solution semigroup for the reactor problem

b. spectral and asymptotic behavior

notes

3. semigroups for second-order cauchy problems

a. the state space x = xb1 × x

b. the state space x = x × x

c. the state space x = xc1 × x

notes

4. semigroups for ordinary differential operators

m. campiti, g. metafune, d. pallara, and s. romanelli

a. nondegenerate operators on r and r+

b. nondegenerate operators on bounded intervals

c. degenerate operators

d. analyticity of degenerate semigroups

notes

5. semigroups for partial differential operators

abdelaziz rhandi

a. notation and preliminary results

b. elliptic differential operators with constant coefficients

c. elliptic differential operators with variable coefficients

notes

6. semigroups for delay differential equations

a. well-posedness of abstract delay differential equations

b. regularity and asymptotics

c. positivity for delay differential equations

notes

7. semigroups for volterra equations

a. mild and classical solutions

b. optimal regularity

c. integro-differential equations

notes

8. semigroups for control theory

a. controllability

b. observability

c. stabilizability and detectability

d. transfer functions and stability

notes

9. semigroups for nonautonomons cauchy problems

roland schnaubelt

a. cauchy problems and evolution families

b. evolution semigroups

c. perturbation theory

d. hyperbolic evolution families in the parabolic case

notes

vii. a brief history of the exponential function

tanja hahn and carla perazzoli

1. a bird's-eye view

2. the functional equation

3. the differential equation

4. the birth of semigroup theory

appendix

a. a reminder of some functional analysis

b. a reminder of some operator theory

c. vector-valued integration

a. the bochner integral

b. the fourier transform

c. the laplace transform

epilogue

determinism: scenes from the interplay between

metaphysics and mathematics

gregor nickel

1. the mathematical structure

2. are relativity, quantum mechanics, and chaos deterministic?

3. determinism in mathematical science from newton to einstein

4. developments in the concept of object from leibniz to kant

5. back to some roots of our problem: motion in history

6. bibliography and further reading

references

list of symbols and abbreviations

index

作者简介

《线性发展方程的单参数半群》是springer数学研究生丛书之一，全面讲述了强连续线性算子的单参半群理论。本书的最大特点是在常微分和偏分方程算子、衰退方程和volterra方程和控制理论中广泛应用。而且，书中也强调了一些哲学动机和历史背景。

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