Gespeichert in:
| Titel: | Differential equations with operator coefficients with applications to boundary value problems for partial differential equations |
|---|---|
| Von: |
Vladimir Kozlov ; Vladimir Maz'ya
|
| Person: |
Kozlov, Vladimir
1954- Verfasser aut Mazʹja, Vladimir Gilelevič 1937- |
| Hauptverfassende: | , |
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1999
|
| Schriftenreihe: | Springer Monographs in mathematics
|
| Schlagworte: | |
| Online-Zugang: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008294187&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | XX, 442 S. |
| ISBN: | 3540651195 |
Internformat
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| 100 | 1 | |a Kozlov, Vladimir |d 1954- |e Verfasser |0 (DE-588)103679014 |4 aut | |
| 245 | 1 | 0 | |a Differential equations with operator coefficients |b with applications to boundary value problems for partial differential equations |c Vladimir Kozlov ; Vladimir Maz'ya |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1999 | |
| 300 | |a XX, 442 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer Monographs in mathematics | |
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| 700 | 1 | |a Mazʹja, Vladimir Gilelevič |d 1937- |e Verfasser |0 (DE-588)121490602 |4 aut | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-008294187 | |
Datensatz im Suchindex
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| DE-BY-UBR_location | UB Lesesaal Mathematik |
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| adam_text |
Table of Contents
Introduction XV
Part I. Differential Equations with Constant Operator Coefficients
1. Power Exponential Zeros 3
1.1 Introduction 3
1.2 Basics on Operator Pencils 4
1.2.1 Notation 4
1.2.2 Decomposition of the Resolvent Near the Pole . 6
1.2.3 Two Term Quadratic Pencils 7
1.3 Power Exponential Solutions of the Homogeneous Equation 10
1.3.1 Notation. Spaces Z(A,Xv) and Z{A*,JV) 10
1.3.2 A Biorthogonality Condition 11
1.3.3 Proof of Proposition 1.3.1 14
1.3.4 Two Term Second Order Equations 14
1.3.5 A Construction of Canonical Systems
of Jordan Chains 16
1.4 Power Exponential Solutions
of the Nonhomogeneous Equation 16
1.5 Applications to Elliptic Partial Differential Equations
with Constant Coefficients 17
1.5.1 Neumann Problem in a Cylinder 17
1.5.2 The Dirichlet Problem in a Cone 18
1.5.3 Properties of the Operator Pencil (1.44) 19
1.5.4 The Adjoint Pencil 22
1.5.5 The Dirichlet Problem in a Half Space 22
1.5.6 Elliptic Equations in R" \ O 23
1.6 Comments 25
2. Differential Operator Equations
in Weighted Sobolev Spaces 27
2.1 Introduction 27
2.2 The Operator Pencil A(\) 27
VI Table of Contents
2.2.1 Conditions on A(X) 27
2.2.2 Examples of Pencils Satisfying Conditions I and II 29
2.2.3 Notation 30
2.3 Some Spaces of Vector Valued Functions 31
2.3.1 Sobolev Spaces 31
2.3.2 Weighted Sobolev Spaces 32
2.4 Solvability in W%(R) 33
2.5 Application to the Dirichlet Problem in a Cylinder 34
2.6 Green's Kernel 36
2.6.1 Definition of Green's Kernel 36
2.6.2 Properties of G(t) 37
2.6.3 Integral Representation of Solutions 38
2.7 Asymptotic Decompositions of Green's Kernel 39
2.7.1 Representations for G(t) 39
2.7.2 Representations for G(t r) 41
2.8 Asymptotics of Solutions in Wj(R) 43
2.8.1 Asymptotic Representations 43
2.8.2 Solutions of the Homogeneous Equation 44
2.9 A Local Estimate for Solutions 45
2.10 Application to the Dirichlet Problem in a Cone 46
2.11 Comments 48
3. Solutions in a Local Sobolev Space 49
3.1 Introduction 49
3.2 Zeros of A(Dt) 50
3.2.1 Uniqueness for Homogeneous Equation 50
3.2.2 Behaviour of Zeros at Infinity 51
3.3 Unique Solvability of the Nonhomogeneous Equation 52
3.3.1 An Auxiliary Existence Result 52
3.3.2 Unique Solvability 54
3.4 Solutions Corresponding to a Strip 56
3.5 Comparison Principle 56
3.5.1 Comparison Equation and Its Green Function . 56
3.5.2 Solvability Criterion for the Comparison Equation 58
3.5.3 Comparison Principle 59
3.5.4 A General Asymptotic Representation
of the (fc_, fc+) Solution 60
3.6 Estimates for Solutions on a Semiaxis 63
3.7 The Phragmen Lindelof Principle 64
3.8 Asymptotics of Solutions Corresponding to a Strip 65
3.8.1 A Representation for the Difference of Two Solutions 65
3.8.2 An Asymptotic Formula 66
3.8.3 Description of Solutions to Homogeneous Equation 67
3.9 Applications to Boundary Value Problems 68
3.9.1 The Dirichlet Problem in a Cylinder 68
Table of Contents VII
3.9.2 The Neumann Problem in a Cylinder 70
3.9.3 The Dirichlet Problem in a Cone 71
3.10 Comments 74
4. Two Weight L2 Estimates 75
4.1 Introduction 75
4.2 Weighted Sobolev Spaces 76
4.3 Uniqueness of Solutions in iy*(R; 7) 77
4.4 Existence of Solutions in We(R; 7) 79
4.4.1 Principal Result 79
4.4.2 Auxiliary Results on Operators
of Multiple Integration 80
4.4.3 Proof of Theorem 4.4.1 84
4.4.4 Power Exponential Weight Functions 85
4.5 Application to the Dirichlet Problem in a Cone 85
4.6 Comments 87
Part II. Differential Equations with Variable Operator Coefficients
5. Existence, Uniqueness and "Pointwise" Estimates 91
5.1 Introduction 91
5.2 Auxiliary Information on the Comparison Equation 93
5.2.1 Green's Function 93
5.2.2 Existence and Uniqueness Results
for the Comparison Equation 96
5.3 Existence 98
5.3.1 Assumptions on the Operator L 98
5.3.2 Construction of a (fc_, | ) Solution 99
5.4 Uniqueness Theorems 101
5.4.1 A Class of Uniqueness 101
5.4.2 Another Class of Uniqueness 102
5.4.3 The Case m± 2 104
5.4.4 An Explicit Uniqueness Condition in Terms of u o . 104
5.5 Behaviour of Zeros at Infinity 104
5.5.1 Zeros of L 104
5.5.2 Zeros of the Adjoint Operator 105
5.6 Estimates for Solutions on the Semiaxis t to 107
5.7 Applications to Partial Differential Equations
with Variable Coefficients 109
5.7.1 The Dirichlet Problem in a Cylinder 109
5.7.2 The Dirichlet Problem in a Cone 110
VIII Table of Contents
6. Corollaries of Previous Results
Under Special Assumptions on L(t, Dt) 113
6.1 Introduction 113
6.2 General Perturbations 114
6.2.1 Existence and Uniqueness Theorems 114
6.2.2 Estimates for Solutions at Infinity 116
6.3 The Case m+ = ro_ = 1 117
6.3.1 Equation (5.1) on R 117
6.3.2 Equation (6.8) on a Semiaxis 120
6.4 Estimates of the Phragmen Lindelof Type
for Solutions of (6.8) when p Dominates Either t~m+
or rm for Large t 0 121
6.5 Applications to Partial Differential Equations in a Cylinder 124
6.6 Other Applications 126
6.6.1 Isolated Singularities 126
6.6.2 The Neumann Problem in a Cylinder 127
6.6.3 The Dirichlet Problem
for Other Nonsmooth Domains 128
6.7 Comments 131
7. Two Weight i2 Estimates for Equations
with Variable Coefficients 133
7.1 Introduction 133
7.2 Uniqueness Theorems in Weighted Sobolev Spaces 135
7.3 Existence Theorems for Solutions in Weighted Sobolev
Spaces and Two Weight Estimates 136
7.4 Unique Solvability in W^_0+ (R) 138
7.5 The Case m± = 1 '. .+. 140
7.6 Two Weight Estimates when p Dominates t~m+ 144
7.7 Comments 146
8. Connection of Solutions Corresponding to Different Strips 147
8.1 Introduction 147
8.2 Auxiliary Information 147
8.2.1 Notation 147
8.2.2 Estimates for Green's Functions
of the Comparison Equations 148
8.2.3 An Auxiliary Existence Result 150
8.3 Zeros of L[t, Dt) 151
8.3.1 The Class X(L) 151
8.3.2 The Dimension of X(L) 152
8.3.3 The Norm in X(L) 154
8.4 Solutions of (5.1) Corresponding to Different Strips 155
8.4.1 The Auxiliary Dual Space 155
8.4.2 TheSubspace£*(L) 156
Table of Contents IX
8.4.3 The Difference of Two Solutions Belongs to X(L) . 157
8.4.4 A Sesquilinear Form and the Dimension of X*(L) . 158
8.4.5 Main Result 159
8.5 Structure of Solutions of (6.8) at Infinity 160
8.5.1 The Spaces Yi(L) and Y2(L) 160
8.5.2 (Yi,K2) Spaces 161
8.5.3 An Asymptotic Representation 162
8.6 Comments 164
9. Applications to the Case of Perturbations
Vanishing at Infinity 165
9.1 Introduction 165
9.2 The Case p(t) »0ast » ±oo 166
9.2.1 Description of the Class X(L) 167
9.2.2 Description of the Class X*(L) 167
9.2.3 Characteristic Exponents 168
9.2.4 The Difference of Two Solutions 168
9.3 The Case p(t) + 0 as t » +oo 169
9.3.1 Structure of Solutions at +oo 169
9.3.2 Existence of Characteristic Exponents
for Solutions of (8.53) 171
9.3.3 Asymptotic Equivalence of Two Equations 172
9.4 The Case of Absence of Generalized Eigenvectors 173
9.4.1 Zeros of L and L* 173
9.4.2 Relation of Solutions Corresponding
to Different Strips 174
9.4.3 An Asymptotic Representation of Solutions at +oo 174
9.4.4 The Asymptotic Equivalence of Two Equations . . 177
9.5 Application to the Local Regularity of Solutions
to Elliptic Equations 178
9.6 Comments 180
10. Variants and Extensions of the Previous Theory 181
10.1 Introduction 181
10.2 Estimates of Solutions to Operator Differential Inequalities 182
10.3 Perturbation of a Differential Operator
with Variable Coefficients 184
10.3.1 General Case 184
10.3.2 A Second Order Differential Operator 185
10.3.3 Perturbations of the Second Order
Differential Operator 188
10.4 Applications to Partial Differential Equations 191
10.5 "Parabolic" First Order Operators
with a Variable Dissipative Term 194
10.5.1 Unperturbed Operator 194
X Table of Contents
10.5.2 Perturbed Operator 196
10.6 "Hyperbolic" Operator Equations
with Constant Coefficients 200
10.6.1 Assumptions on the Pencil A(\) 200
10.6.2 Example of a Second Order Differential Operator . 201
10.6.3 Solutions in Weighted Sobolev Spaces 202
10.6.4 Comparison Principle and Solvability in Wf~c1(R) . 205
10.7 "Hyperbolic" Operator Equation with Variable Coefficients 208
10.8 The Operator L in Variational Form 209
10.8.1 Function Spaces 209
10.8.2 An Equivalent Norm in W~q(R) 214
10.8.3 Assumptions on the Pencil A(\) 216
10.8.4 Equation with Constant Coefficients 218
10.8.5 Equation with Variable Coefficients 220
10.8.6 The Variational Form of the Dirichlet Problem
in a Cone 222
10.9 Ordinary Differential Equations in Banach Spaces 226
10.9.1 Assumptions on the Operator
with Constant Coefficients 226
10.9.2 Solvability of the Equation
with Constant Coefficients 229
10.9.3 Uniqueness for the Equation
with Constant Coefficients 232
10.9.4 Comparison Principle for the Equation
with Constant Coefficients 234
10.9.5 Equation with Variable Coefficients 234
10.9.6 Application to the Dirichlet Problem in a Cylinder 235
10.10 Applications to Elliptic Boundary Value Problems in a Cone 237
10.10.1 The Comparison Principle for the Model Problem . 237
10.10.2 The Comparison Principle for the Boundary Value
Problem with Variable Coefficients 240
Part III. Asymptotic Theory of Operator Differential Equations
11. Complete Asymptotic Expansions Under Exponential
and Power Perturbations of A(Dt) 245
11.1 Introduction 245
11.2 Perturbation with Exponential Decay
(Homogeneous Equation) 246
11.3 Perturbation with Exponential Decay
(Nonhomogeneous Equation) 249
11.4 Perturbation in the Form of Laurent Series
(Homogeneous Equation) 251
Table of Contents XI
11.5 Perturbation in the Form of Laurent Series
(Nonhomogeneous Equation) 255
11.6 The Dirichlet Problem for a Cuspidal Domain 259
11.7 Comments 262
12. Reduction to a First Order System 265
12.1 Introduction 265
12.2 Prerequisites for the Subsequent Asymptotic Theory 265
12.3 Linearization of the Pencil ^4(A) 268
12.4 Canonical Sets of Jordan Chains of IeA + 21 271
12.5 The Riesz Projector and Its Properties 273
12.6 The Vector Function Spaces §ioc(R), Xioc(R) and Yioc(R) . 275
12.7 Existence of Solutions in Xloc(l) 277
12.8 Uniqueness of Solutions in Xioc(M) 279
12.9 From the Equation with Variable Coefficients to a First
Order System 281
12.10 Comments 283
13. General Asymptotic Representation 285
13.1 Introduction 285
13.2 Spectral Decomposition of the First Order System 286
13.3 Solvability of the Infinite Dimensional Part
of the Split System 287
13.4 Uniqueness for the Infinite Dimensional Part
of the Split System 290
13.5 The Function a 291
13.6 Reduction of the Split System
to a Finite Dimensional System 294
13.7 Estimates for the Operator K and the Function f 297
13.8 Estimates for the Function 0 298
13.9 The Finite Dimensional System in the Matrix Form 302
13.10 Operator Differential Equations on a Semiaxis 305
13.10.1 Preliminaries 305
13.10.2 An Existence Result 306
13.10.3 A Representation for Solutions to (13.92) 307
13.10.4 The Matrix Form of System (13.95) 308
13.11 Properties of Zeros of the Finite Dimensional System 309
13.11.1 The System on R 309
13.11.2 The System on a Semiaxis 311
13.12 Estimate for a Neumann Series 312
14. Power Exponential Asymptotics 317
14.1 Introduction 317
14.2 Special Solutions of the Finite Dimensional System 320
14.2.1 The Functions a, 0, 7 and \ 320
XII Table of Contents
14.2.2 Lemma on Special Solutions 321
14.3 Special Solutions of (14.1) 325
14.3.1 Construction of Solutions
with Prescribed Asymptotics 325
14.3.2 Main Result 327
14.4 Asymptotics of Arbitrary Solutions of (14.1) 330
14.4.1 Main Result 330
14.4.2 Refinement of the Asymptotics 331
14.5 Nonhomogeneous Finite Dimensional System 334
14.6 A Special Solution of the Nonhomogeneous System (14.2) . 336
14.7 Asymptotics of Solutions
to the Nonhomogeneous System (14.2) 338
14.8 Asymptotics of Solutions in a Weighted Sobolev Space . 340
14.9 Asymptotic Behaviour of Solutions to Elliptic Equations
near an Interior Point 342
14.10 Comments 344
15. The Case of One Simple Eigenvalue on the Line 345
15.1 Introduction 345
15.2 A Comparison Principle for Integral Inequalities 346
15.3 Estimate for Solutions of the Scalar Equation (13.108) 347
15.4 Zeros with Prescribed Asymptotics at Infinity 353
15.5 Asymptotics of Solutions of the Homogeneous
Higher Order Equation at +oo 356
15.6 Asymptotics of Solutions to the Nonhomogeneous Equation 361
15.7 Comments 366
16. Several Simple Eigenvalues on the Line 369
16.1 Introduction 369
16.2 Special Solutions of the Finite Dimensional System 369
16.2.1 Functions a and 0 369
16.2.2 Dichotomy Condition 370
16.2.3 Special Solutions of the Finite Dimensional System
(16.3) 371
16.2.4 Proof of Lemma 16.2.2 372
16.3 Asymptotic Formulae for Solutions 376
16.4 A Second Order Equation 379
16.5 Application to the Schrodinger Equation in a Cylinder . 382
16.6 Comments 383
17. The Case of a Single Multiple Eigenvalue 385
17.1 Introduction 385
17.2 Solution of an Auxiliary Matrix Equation 386
17.3 Special Solutions of the Finite Dimensional System 390
17.3.1 Prerequisites 390
Table of Contents XIII
17.3.2 Lemma on Special Solutions 392
17.4 Asymptotics of Solutions 397
17.5 An Example 399
17.6 Comments 402
A. Holomorphic Operator Functions 403
A.I Introduction 403
A.2 Prerequisites on Fredholm Operators 404
A.3 Basic Notions of the Spectral Theory
of Holomorphic Operator Functions 405
A.4 Canonical Generating System in S(F, Xq)
and Canonical Set of Jordan Chains 407
A.5 The Local Equivalence of Holomorphic Operator Functions 411
A.6 The Smith Form of a Holomorphic Matrix Function 413
A.7 The Resolvent of a Holomorphic Matrix Function 417
A.8 Fredholm Holomorphic Operator Functions 419
A.9 The Adjoint Holomorphic Operator Function 422
A.10 The Structure of F(X) 1 Near the Pole 425
References 431
Index of Notation 435
Index 439
Index of Names 441 |
| any_adam_object | 1 |
| author | Kozlov, Vladimir 1954- Mazʹja, Vladimir Gilelevič 1937- |
| author_GND | (DE-588)103679014 (DE-588)121490602 |
| author_facet | Kozlov, Vladimir 1954- Mazʹja, Vladimir Gilelevič 1937- |
| author_role | aut aut |
| author_sort | Kozlov, Vladimir 1954- |
| author_variant | v k vk v g m vg vgm |
| building | Verbundindex |
| bvnumber | BV012240125 |
| classification_rvk | SK 540 SK 620 SK 920 |
| classification_tum | MAT 340f |
| ctrlnum | (OCoLC)845147048 (DE-599)BVBBV012240125 |
| dewey-full | 515.35 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.35 |
| dewey-search | 515.35 |
| dewey-sort | 3515.35 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012240125 |
| illustrated | Not Illustrated |
| indexdate | 2025-04-15T08:00:58Z |
| institution | BVB |
| isbn | 3540651195 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008294187 |
| oclc_num | 845147048 |
| open_access_boolean | |
| owner | DE-20 DE-703 DE-824 DE-19 DE-BY-UBM DE-898 DE-BY-UBR DE-29T DE-91G DE-BY-TUM DE-384 DE-355 DE-BY-UBR DE-706 DE-526 DE-634 DE-83 DE-11 DE-188 |
| owner_facet | DE-20 DE-703 DE-824 DE-19 DE-BY-UBM DE-898 DE-BY-UBR DE-29T DE-91G DE-BY-TUM DE-384 DE-355 DE-BY-UBR DE-706 DE-526 DE-634 DE-83 DE-11 DE-188 |
| physical | XX, 442 S. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer Monographs in mathematics |
| spellingShingle | Kozlov, Vladimir 1954- Mazʹja, Vladimir Gilelevič 1937- Differential equations with operator coefficients with applications to boundary value problems for partial differential equations Differentialoperator (DE-588)4012251-7 gnd Operatorgleichung (DE-588)4043601-9 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4012251-7 (DE-588)4043601-9 (DE-588)4020929-5 |
| title | Differential equations with operator coefficients with applications to boundary value problems for partial differential equations |
| title_auth | Differential equations with operator coefficients with applications to boundary value problems for partial differential equations |
| title_exact_search | Differential equations with operator coefficients with applications to boundary value problems for partial differential equations |
| title_full | Differential equations with operator coefficients with applications to boundary value problems for partial differential equations Vladimir Kozlov ; Vladimir Maz'ya |
| title_fullStr | Differential equations with operator coefficients with applications to boundary value problems for partial differential equations Vladimir Kozlov ; Vladimir Maz'ya |
| title_full_unstemmed | Differential equations with operator coefficients with applications to boundary value problems for partial differential equations Vladimir Kozlov ; Vladimir Maz'ya |
| title_short | Differential equations with operator coefficients |
| title_sort | differential equations with operator coefficients with applications to boundary value problems for partial differential equations |
| title_sub | with applications to boundary value problems for partial differential equations |
| topic | Differentialoperator (DE-588)4012251-7 gnd Operatorgleichung (DE-588)4043601-9 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Differentialoperator Operatorgleichung Gewöhnliche Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008294187&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kozlovvladimir differentialequationswithoperatorcoefficientswithapplicationstoboundaryvalueproblemsforpartialdifferentialequations AT mazʹjavladimirgilelevic differentialequationswithoperatorcoefficientswithapplicationstoboundaryvalueproblemsforpartialdifferentialequations |
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