Applied semigroups and evolution equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Clarendon Press
1979
|
| Schriftenreihe: | Oxford mathematical monographs
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 387 S. |
| ISBN: | 0198535295 |
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| 245 | 1 | 0 | |a Applied semigroups and evolution equations |c Aldo Belleni-Morante |
| 264 | 1 | |a Oxford |b Clarendon Press |c 1979 | |
| 300 | |a XV, 387 S. | ||
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| 490 | 0 | |a Oxford mathematical monographs | |
| 650 | 7 | |a Equations d'évolution |2 ram | |
| 650 | 4 | |a Evolution equations | |
| 650 | 4 | |a Semigroups of operators | |
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Datensatz im Suchindex
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| adam_text | CONTENTS
INTRODUCTION 1
1. BANACH AND HILBliRT SPACES
1.1. Banach and Hilbert spaces 7
1.2. Examples of Banach and Hilbert spaces 12
Example 1.1./? 12
Example 1.2. lp, p 1 13
Example 1.3. C([a,b]), °° a b +°° 15
Example 1.4. Lp (a ,b) , p 1, » a i s +°° 16
Example 1.5. Closed linear subsets of a 20
B space X
Example 1.6. X = JjX^, 21
1.3. Generalized derivatives 21
Example 1.7. A function belonging to C^( 2,+2) 22
Example 1.8. The generalized derivative of 23
fix) = x
1.4. Sobolev spaces of integer order 25
Example 1.9. W™ P(Q) ? f/ p(fi) ¦+ Lp {Q) 29
Example 1.10. Lv {9.) * Lp{n) , 1 p p ¦» 30
Example 1.11. W1 2^1) • CgC/?1) 31
1.5. Sobolev spaces of fractional order 32
Exercises 35
2. OPERATORS IN BANACH SPACES
2.1. Notation and basis definitions 38
Example 2.1. The operator B(f) = f2 and its 38
inverse
2.2. Bounded linear operators 39
2.3. Examples of bounded linear operators 44
Example 2.2. Operators on nn 44
Example 2.3. Operators on I 45
Example 2.4. The Fredholm integral operator 46
on L2(a,b)
Example 2.5. The operator A = d/da:, with 47
D(A) = W1 2^1) and j?U)c I2^1)
x CONTENTS
Example 2.6. Extension of a densely defined 48
bounded operator
Example 2.7. Linear functionals on L (a ,b) 50
2.4. Lipschitz operators 51
Example 2.8. The operator A (f) = f with 52
flU) = {f.feC{[a,b]) ; II f II r}
and i?0O c C( [a ,b])
Example 2.9. The operator A(/) = y/ /(1+/2) 52
with Z?04) = L2(a,b) ,
R{A) c L2(a,fc)
Example 2.10. The operator ,4 (f) = f2 with 53
DU) = {f.feW1 2^1) ;
H7 /nlh ? £ r} and ^f 4) c
Example 2.11. The operator F(/) = (Af)(Bf) 56
with .4 and B belonging to
*U,C([a,i]))
Example 2.12. Contraction mapping theorem 57
2.5. Closed operators 60
2.6. Self adjoint operators 63
Example 2.13. The Fredholm integral operator 68
Example 2.14. The heat diffusion operator 68
Af = kf with D(A) = {f:feL2{.a,b)
f eL2(a,b );f(a) = /(2 ) = Q}
Example 2.15. The operator Af = ?~Y [ ky Vf ] , 70
with DU) = IS + 2 2C«1) ,
2.7. Spectral properties: basic definitions 73
2.8. Spectral properties: examples 77
Example 2.16. Resolvent of Ae (X) 77
Example 2.17. Spectral properties of an 79
operator in I
Example 2.18. The convection operator 83
A = ud/da; with
D(A) = {f:f€L2(_a,b) ;f eL2(a,b) ;
f(a) = 0}, flU) c L2(a,b)
Example 2.19. The heat diffusion operator 87
Example 2.20. Resolvent of A+B with AeV(.X) 94
and Be (X)
CONTENTS xi
Example 2.21. Resolvent set of a self adjoint 95
operator
Exercises 98
3. ANALYSIS IN BANACH SPACES
3.1. Strong continuity 101
Example 3.1. Strong continuity in Rn 102
Example 3.2. Strong continuity in I 102
Example 3.3. Strong continuity in C{[a,b}) 103
Example 3.4. Strong continuity in L (a ,b) 104
Example 3.5. Strong continuity in %(X,Y) 104
3.2. Strong derivative 106
Example 3.6. Strong derivative in C([a,b]) 107
3.3. Strong Riemann integral 110
Example 3.7. The operator Aw = dw/dt with 113
D{A) = {w:weC([0,tQ];X);
dw/dteC{[O,to);X); w(0) = 9^},
R(A) c C([0,tQ];X)
Example 3.8. Differentiation under the 114
integral sign
3.4. The differential equation dw/dt = F(u) 114
3.5. Holomorphic functions 120
Example 3.9. The resolvent of a closed operator
operator 122
3.6. Frechet derivative 123
Example 3.10. F derivative of F(f) = fQ+Bf 125
with Be %(X)
Example 3.11. F derivative of F(f) = f with 125
D{F) = L2(a,b), R(F) c L1(a,Z3)
Exercises 126
4. SEMIGROUPS
4.1. Linear initial value problems 128
4.2. The case Ae3S{X) 130
Example 4.1. Ae3}{Rn) 135
Example 4.2. An integro differential system 136
Example 4.3. Best approximation of a non 138
linear problem by means of a
linear problem
4.3. The case Ae^(X) 139
xil CONTENTS :
Example 4.4. The convection operator 141
Example 4.5. The heat diffusion operator 141
Example 4.6. An operator in I 141
Example 4.7. The case Ae es{X) 146
Example 4.8. The case A , = iA 146
with A self adjoint
4.4. The case Ae^(l,0;X): two preliminary lemmas 146
4.5. The semigroup generated by Ae $(l,0;X) 152
4.6. The cases Ae ${M,0 ;X) , S?(M,g;J), g (M, ;X) 158
Example 4.9. Conservation of the norm of 161
u(t) = expO^MQ with A^e g (l,0;X)
4.7. The homogeneous and the non homogeneous 162
initial value problems
Example 4.10. Discretization of the time 164
like variable
Example 4.11. Oscillating sources 171
Example 4.12. Periodic solutions 172
Example 4.13. Oscillating heat sources 175
Exercises 176
5. PERTURBATION THEOREMS
5.1. Introduction 178
5.2. Bounded perturbations 179
Example 5.1. An integral perturbation of the 182
heat diffusion operator
Example 5.2. Ae g(l,0 ;ll) , BeMil1) 184
Example 5.3. Ae S{.U, 6 ;X) , B = zB Q with 184
Bq£@{x) and zet
5.3. The cases B = B(t)e. %(x) and B relatively 187
bounded
Example 5.4. The integral operator B(t) of 189
Exercise 3.3
Example 5.5. Relative boundedness of the con 190
vection operator with respect
to the heat diffusion operator
5.4. The semilinear case 191
Example 5.6. The non linear temperature 192
dependent source F(t) =
yT2(x;t){l + T2(a;;t)} 1
CONTENTS xiii
5.5. Global solution of the semilinear problem 205
(5.31)
Example 5.7. The case Re O (/),/) All/II2 211
VfeX, with X = a Hilbert space
Example 5.8. F(f) = (1+/2) 1, F(f) 212
= ^[l+Z2] 1 with DO) = D(F)
= C{[a,b])
Example 5.9. F(f) = (Af)(Bf) with A and B 214
belonging to .#(C([a,fc]))
Example 5.10. A linearization procedure 215
Exercises 217
6. SEQUENCES OF SEMIGROUPS
6.1. Sequences of semigroups exp(tA.)e (X) 219
Example 6.1. exp (tA .) f *• exp (tA) f with A .e M{X) 223
6.2. Sequences of Banach spaces 225
Example 6.2. A sequence of B spaces approxi 225
mating C([a ,b])
Example 6.3. A sequence of B spaces approxi 227
mating L (a,b)
6.3. Sequences of semigroups exp(t4 .) e.%(X .) 228
Example 6.4. Galerkin method 233
Example 6.5. Discretization of the operator 237
ud/dx
Exericses 241
7. SPECTRAL REPRESENTATION OF CLOSED OPERATORS
AND OF SEMIGROUPS
7.1. Introduction 244
7.2. Projections 245
Example 7.1. Projections in Rn 247
Example 7.2. Projections on a subspace of 248
a Hilbert space
Example 7.3. A projection operator in L 248
7.3. Isolated points of the spectrum of Ae^(X) 249
7.4. Laurent expansions of R(z,A) 253
7.5. Isolated eigenvalues 256
Example 7.4. Spectral properties of an 258
2
operator in (E
xiv CONTENTS
Example 7.5. Spectral properties of an operator
in I1 262
Example 7.6. Spectral properties of the
heat diffusion operator 263
7.6. Spectral representation of A and of exp(tA) 265
Example 7.7. Spectral representation of 271
exp(t^) with Aei2( L2)
Example 7.8. Spectral representation of 271
exp(t^l) with 4eg?(l,0;Z )
Example 7.9. Spectral representation of the 273
heat diffusion operator
Example 7.10. Relationships between °(A) and 273
a(exp(*4))
Exercises 277
8. HEAT CONDUCTION IN RIGID BODIES AND SIMILAR PROBLEMS
8.1. Introduction 280
8.2. A linear heat conduction problem 281
8.3. A semilinear heat conduction problem 284
8.4. Positive solutions 288
Exercises 291
9. NEUTRON TRANSPORT
9.1. Introduction 295
9.2. Linear neutron transport in L 297
9.3. Spectral properties of the transport operator 304
9.4. A semilinear neutron transport problem 308
Exercises 316
10. A SEMILINEAR PROBLEM FROM KINETIC THEORY OF
VEHICULAR TRAFFIC
10.1. Introduction 319
10.2. Preliminary lemmas 321
10.3. The operators F, K ., and #2 326
10.4. The operators J and K^ 328
10.5. Global solution of the abstract problem 331
(10.11)
Exercises 335
CONTENTS xv
11. THE TELEGRAPHIC EQUATION AND THE WAVE EQUATION
11.1. Introduction 338
11.2. Preliminary lemmas 340
11.3. The abstract version of the telegraphic 348
system (11.4)
11.4. The telegraphic equation and the wave equation 350
Exercises 355
12. A PROBLEM FROM QUANTUM MECHANICS
12.1. Introduction 357
12.2. Spectral properties of iA 358
12.3. Bounded perturbations 360
Exercises 363
13. A PROBLEM FROM STOCHASTIC POPULATION THEORY
13.1. Introduction 365
13.2. The abstract problem 366
13.3. Preliminary lemmas 367
13.4. Strict solution of the approximating problem 369
(13.7)
13.5. A property of the strict solution of the 371
approximating problem
13.6. Strict solution of problem (13.6) 375
13.7. The equation for the first moment (n)(t) 377
of the bacteria population
Exercises 381
BIBLIOGRAPHY 383
Subject index 385
|
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| id | DE-604.BV002259199 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:42:56Z |
| institution | BVB |
| isbn | 0198535295 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001484668 |
| oclc_num | 6489994 |
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| physical | XV, 387 S. |
| publishDate | 1979 |
| publishDateSearch | 1979 |
| publishDateSort | 1979 |
| publisher | Clarendon Press |
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| spelling | Belleni-Morante, Aldo Verfasser aut Applied semigroups and evolution equations Aldo Belleni-Morante Oxford Clarendon Press 1979 XV, 387 S. txt rdacontent n rdamedia nc rdacarrier Oxford mathematical monographs Equations d'évolution ram Evolution equations Semigroups of operators Anwendung (DE-588)4196864-5 gnd rswk-swf Evolutionsgleichung (DE-588)4129061-6 gnd rswk-swf Halbgruppe (DE-588)4022990-7 gnd rswk-swf Halbgruppe (DE-588)4022990-7 s Anwendung (DE-588)4196864-5 s DE-604 Evolutionsgleichung (DE-588)4129061-6 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001484668&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Belleni-Morante, Aldo Applied semigroups and evolution equations Equations d'évolution ram Evolution equations Semigroups of operators Anwendung (DE-588)4196864-5 gnd Evolutionsgleichung (DE-588)4129061-6 gnd Halbgruppe (DE-588)4022990-7 gnd |
| subject_GND | (DE-588)4196864-5 (DE-588)4129061-6 (DE-588)4022990-7 |
| title | Applied semigroups and evolution equations |
| title_auth | Applied semigroups and evolution equations |
| title_exact_search | Applied semigroups and evolution equations |
| title_full | Applied semigroups and evolution equations Aldo Belleni-Morante |
| title_fullStr | Applied semigroups and evolution equations Aldo Belleni-Morante |
| title_full_unstemmed | Applied semigroups and evolution equations Aldo Belleni-Morante |
| title_short | Applied semigroups and evolution equations |
| title_sort | applied semigroups and evolution equations |
| topic | Equations d'évolution ram Evolution equations Semigroups of operators Anwendung (DE-588)4196864-5 gnd Evolutionsgleichung (DE-588)4129061-6 gnd Halbgruppe (DE-588)4022990-7 gnd |
| topic_facet | Equations d'évolution Evolution equations Semigroups of operators Anwendung Evolutionsgleichung Halbgruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001484668&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bellenimorantealdo appliedsemigroupsandevolutionequations |