Asymptotic theory of elliptic boundary value problems in singularly perturbed domains: 2
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English German |
| Published: |
Basel [u.a.]
Birkhäuser
(2000)
|
| Series: | Operator theory
112 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIII, 323 S. |
| ISBN: | 3764363983 |
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| 100 | 1 | |a Mazʹja, Vladimir Gilelevič |d 1937- |e Verfasser |0 (DE-588)121490602 |4 aut | |
| 245 | 1 | 0 | |a Asymptotic theory of elliptic boundary value problems in singularly perturbed domains |n 2 |c Vladimir Maz'ya ; Serguei Nazarov ; Boris Plamenevskij. Transl. from the German by Georg Heinig and Christian Posthoff |
| 264 | 1 | |c (2000) | |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser | |
| 300 | |a XXIII, 323 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Operator theory |v 112 | |
| 700 | 1 | |a Nazarov, Sergej A. |e Verfasser |4 aut | |
| 700 | 1 | |a Plamenevskij, Boris A. |e Verfasser |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV013152946 |g 2 |
| 830 | 0 | |a Operator theory |v 112 |w (DE-604)BV000000970 |9 112 | |
| 856 | 4 | 2 | |m DNB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008961722&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-008961722 | |
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CONTENTS
VOLUME
I
PREFACE
.
XXI
PART
I
BOUNDARY
VALUE
PROBLEMS
FOR
THE
LAPLACE
OPERATOR
IN
DOMAINS
PERTURBED
NEAR
ISOLATED
SINGULARITIES
CHAPTER
1
DIRICHLET
AND
NEUMANN
PROBLEMS
FOR
THE
LAPLACE
OPERATOR
IN
DOMAINS
WITH
CORNERS
AND
CONE
VERTICES
1.1
BOUNDARY
VALUE
PROBLEMS
FOR
THE
LAPLACE
OPERATOR
IN
A
STRIP
.
3
1.1.1
THE
DIRICHLET
PROBLEM
.
3
1.1.2
THE
COMPLEX
FOURIER
TRANSFORM
.
5
1.1.3
ASYMPTOTICS
OF
SOLUTION
OF
THE
DIRICHLET
PROBLEM
.
6
1.1.4
THE
NEUMANN
PROBLEM
.
7
1.1.5
FINAL
REMARKS
.
8
1.2
BOUNDARY
VALUE
PROBLEMS
FOR
THE
LAPLACE
OPERATOR
IN
A
SECTOR
.
8
1.2.1
RELATIONSHIP
BETWEEN
THE
BOUNDARY
VALUE
PROBLEMS
IN
A
SECTOR
AND
A
STRIP
.
8
1.2.2
THE
DIRICHLET
PROBLEM
.
10
1.2.3
THE
NEUMANN
PROBLEM
.
10
1.3
THE
DIRICHLET
PROBLEM
IN
A
BOUNDED
DOMAIN
WITH
CORNER
.
11
1.3.1
SOLVABILITY
OF
THE
BOUNDARY
VALUE
PROBLEM
.
11
1.3.2
PARTICULAR
SOLUTIONS
OF
THE
HOMOGENEOUS
PROBLEM
.
13
1.3.3
ASYMPTOTICS
OF
SOLUTION
.
15
1.3.4
A
DOMAIN
WITH
A
CORNER
OUTLET
TO
INFINITY
.
17
1.3.5
ASYMPTOTICS
OF
THE
SOLUTIONS
FOR
PARTICULAR
RIGHT-HAND
SIDES
.
18
1.3.6
THE
DIRICHLET
PROBLEM
FOR
THE
OPERATOR
YY
-
1
.
22
1.3.7
THE
DIRICHLET
PROBLEM
IN
A
DOMAIN
WITH
PIECEWISE
SMOOTH
BOUNDARY
.
26
1.4
THE
NEUMANN
PROBLEM
IN
A
BOUNDED
DOMAIN
WITH
A
CORNER
.
30
1.5
BOUNDARY
VALUE
PROBLEMS
FOR
THE
LAPLACE
OPERATOR
IN
A
PUNCTURED
DOMAIN
AND
THE
EXTERIOR
OF
A
BOUNDED
PLANAR
DOMAIN
.
34
1.5.1
DIRICHLET
AND
NEUMANN
PROBLEMS
IN
A
PUNCTURED
PLANAR
DOMAIN
.
34
1.5.2
BOUNDARY
VALUE
PROBLEMS
IN
THE
EXTERIOR
OF
A
BOUNDED
DOMAIN
.
36
VI
CONTENTS
1.6
BOUNDARY
VALUE
PROBLEMS
IN
MULTI-DIMENSIONAL
DOMAINS
.
37
1.6.1
A
DOMAIN
WITH
A
CONICAL
POINT
.
37
1.6.2
A
PUNCTURED
DOMAIN
.
39
1.6.3
BOUNDARY
VALUE
PROBLEMS
IN
THE
EXTERIOR
OF
A
BOUNDED
DOMAIN
.
40
CHAPTER
2
DIRICHLET
AND
NEUMANN
PROBLEMS
IN
DOMAINS
WITH
SINGULARLY
PERTURBED
BOUNDARIES
2.1
THE
DIRICHLET
PROBLEM
FOR
THE
LAPLACE
OPERATOR
IN
A
THREE-DIMENSIONAL
DOMAIN
WITH
SMALL
HOLE
.
44
2.1.1
DOMAINS
AND
BOUNDARY
VALUE
PROBLEMS
.
44
2.1.2
ASYMPTOTICS
OF
THE
SOLUTION.
THE
METHOD
OF
COMPOUND
EXPANSIONS
.
45
2.1.3
ASYMPTOTICS
OF
THE
SOLUTION.
THE
METHOD
OF
MATCHED
EXPANSIONS
.
47
2.1.4
COMPARISON
OF
ASYMPTOTIC
REPRESENTATIONS
.
51
2.2
THE
DIRICHLET
PROBLEM
FOR
THE
OPERATOR
A
-
1
IN
A
THREE-DIMENSIONAL
DOMAIN
WITH
A
SMALL
HOLE
.
52
2.3
MIXED
BOUNDARY
VALUE
PROBLEMS
FOR
THE
LAPLACE
OPERATOR
IN
A
THREE-DIMENSIONAL
DOMAIN
WITH
A
SMALL
HOLE
.
54
2.3.1
THE
BOUNDARY
VALUE
PROBLEM
WITH
DIRICHLET
CONDITION
AT
THE
BOUNDARY
OF
THE
HOLE
.
54
2.3.2
FIRST
VERSION
OF
THE
CONSTRUCTION
OF
ASYMPTOTICS
.
55
2.3.3
SECOND
VERSION
OF
THE
CONSTRUCTION
OF
ASYMPTOTICS
.
57
2.3.4
THE
BOUNDARY
VALUE
PROBLEM
WITH
THE
NEUMANN
CONDITION
AT
THE
BOUNDARY
OF
THE
GAP
.
59
2.4
BOUNDARY
VALUE
PROBLEMS
FOR
THE
LAPLACE
OPERATOR
IN
A
PLANAR
DOMAIN
WITH
A
SMALL
HOLE
.
59
2.4.1
DIRICHLET
PROBLEM
.
60
2.4.2
MIXED
BOUNDARY
VALUE
PROBLEMS
.
64
2.5
THE
DIRICHLET
PROBLEM
FOR
THE
OPERATOR
A
-
1
IN
A
DOMAIN
PERTURBED
NEAR
A
VERTEX
.
67
2.5.1
FORMULATION
OF
THE
PROBLEM
.
67
2.5.2
THE
FIRST
TERMS
OF
THE
ASYMPTOTICS
.
67
2.5.3
ADMISSIBLE
SERIES
.
70
2.5.4
REDISTRIBUTION
OF
DISCREPANCIES
.
71
2.5.5
THE
SET
OF
EXPONENTS
IN
THE
POWERS
OF
E,
R,
AND
P
.
72
CONTENTS
VII
PART
II
GENERAL
ELLIPTIC
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
PERTURBED
NEAR
ISOLATED
SINGULARITIES
OF
THE
BOUNDARY
CHAPTER
3
ELLIPTIC
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
WITH
SMOOTH
BOUNDARIES,
IN
A
CYLINDER,
AND
IN
DOMAINS
WITH
CONE
VERTICES
3.1
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
WITH
SMOOTH
BOUNDARIES
.
79
3.1.1
THE
OPERATOR
OF
AN
ELLIPTIC
BOUNDARY
VALUE
PROBLEM
.
79
3.1.2
ELLIPTIC
BOUNDARY
VALUE
PROBLEMS
IN
SOBOLEV
AND
HOLDER
SPACES
.
80
3.1.3
THE
ADJOINT
BOUNDARY
VALUE
PROBLEM
(THE
CASE
OF
NORMAL
BOUNDARY
CONDITIONS)
.
83
3.1.4
ADJOINT
OPERATOR
IN
SPACES
OF
DISTRIBUTIONS
.
84
3.1.5
ELLIPTIC
BOUNDARY
VALUE
PROBLEMS
DEPENDING
ON
A
COMPLEX
PARAMETER
.
85
3.1.6
BOUNDARY
VALUE
PROBLEMS
FOR
ELLIPTIC
SYSTEMS
.
89
3.2
BOUNDARY
VALUE
PROBLEMS
IN
CYLINDERS
AND
CONES
.
92
3.2.1
SOLVABILITY
OF
BOUNDARY
VALUE
PROBLEMS
IN
CYLINDERS:
THE
CASE
OF
COEFFICIENTS
INDEPENDENT
OF
T
.
92
3.2.2
ASYMPTOTICS
AT
INFINITY
OF
SOLUTIONS
TO
BOUNDARY
VALUE
PROBLEMS
IN
CYLINDERS
WITH
COEFFICIENTS
INDEPENDENT
OF
T
.
95
3.2.3
SOLVABILITY
OF
BOUNDARY
VALUE
PROBLEMS
IN
A
CONE
.
97
3.2.4
ASYMPTOTICS
OF
THE
SOLUTIONS
AT
INFINITY
AND
NEAR
THE
VERTEX
OF
A
CONE
FOR
BOUNDARY
VALUE
PROBLEMS
WITH
COEFFICIENTS
INDEPENDENT
OF
R
.
99
3.2.5
BOUNDARY
VALUE
PROBLEMS
FOR
ELLIPTIC
SYSTEMS
IN
A
CONE
.
100
3.2.6
ASYMPTOTICS
OF
THE
SOLUTION
FOR
THE
RIGHT-HAND
SIDE
GIVEN
BY
AN
ASYMPTOTIC
EXPANSION
.
104
3.3
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
WITH
CONE
VERTICES
.
106
3.3.1
STATEMENT
OF
THE
PROBLEM
.
106
3.3.2
ASYMPTOTICS
OF
THE
SOLUTION
NEAR
A
CONE
VERTEX
.
107
3.3.3
FORMULAS
FOR
COEFFICIENTS
IN
THE
ASYMPTOTICS
OF
SOLUTION
(UNDER
SIMPLIFIED
ASSUMPTIONS)
.
109
3.3.4
FORMULA
FOR
COEFFICIENTS
IN
THE
ASYMPTOTICS
OF
SOLUTION
(GENERAL
CASE)
.
110
3.3.5
INDEX
OF
THE
BOUNDARY
VALUE
PROBLEM
.
113
CHAPTER
4
ASYMPTOTICS
OF
SOLUTIONS
TO
GENERAL
ELLIPTIC
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
PERTURBED
NEAR
CONE
VERTICES
4.1
FORMULATION
OF
THE
BOUNDARY
VALUE
PROBLEMS
AND
SOME
PRELIMINARY
CONSIDERATIONS
.
115
4.1.1
THE
DOMAINS
.
115
4.1.2
ADMISSIBLE
SCALAR
DIFFERENTIAL
OPERATORS
.
116
4.1.3
LIMIT
OPERATORS
.
117
4.1.4
MATRICES
OF
DIFFERENTIAL
OPERATORS
.
118
4.1.5
BOUNDARY
VALUE
PROBLEMS
.
118
4.1.6
FUNCTION
SPACES
WITH
NORMS
DEPENDING
ON
THE
PARAMETER
E
.
118
VIII
CONTENTS
4.2
TRANSFORMATION
OF
THE
PERTURBED
BOUNDARY
VALUE
PROBLEM
INTO
A
SYSTEM
OF
EQUATIONS
AND
A
THEOREM
ABOUT
THE
INDEX
.
120
4.2.1
THE
LIMIT
OPERATOR
.
120
4.2.2
REDUCTION
OF
THE
PROBLEM
TO
A
SYSTEM
.
121
4.2.3
RECONSTRUCTION
OF
THE
ORIGINAL
PROBLEM
FROM
THE
SYSTEM
.
124
4.2.4
FREDHOLM
PROPERTY
FOR
THE
OPERATOR
OF
THE
BOUNDARY
VALUE
PROBLEM
IN
A
DOMAIN
WITH
SINGULARLY
PERTURBED
BOUNDARY
.
127
4.2.5
ON
THE
INDEX
OF
THE
ORIGINAL
PROBLEM
.
127
4.3
ASYMPTOTIC
EXPANSIONS
OF
DATA
IN
THE
BOUNDARY
VALUE
PROBLEM
.
130
4.3.1
ASYMPTOTIC
EXPANSION
OF
THE
COEFFICIENTS
AND
THE
RIGHT-HAND
SIDES
.
131
4.3.2
ASYMPTOTIC
FORMULAS
FOR
SOLUTIONS
OF
THE
LIMIT
PROBLEMS
.
132
4.3.3
ASYMPTOTIC
EXPANSIONS
OF
OPERATORS
OF
THE
BOUNDARY
VALUE
PROBLEM
.
133
4.3.4
PRELIMINARY
DESCRIPTION
OF
ALGORITHM
FOR
CONSTRUCTION
OF
THE
ASYMPTOTICS
OF
SOLUTIONS
.
134
4.3.5
THE
SET
OF
EXPONENTS
IN
ASYMPTOTICS
OF
SOLUTIONS
OF
THE
LIMIT
PROBLEMS
.
137
4.3.6
FORMAL
EXPANSION
FOR
THE
OPERATOR
IN
POWERS
OF
SMALL
PARAMETER
.
138
4.4
CONSTRUCTION
AND
JUSTIFICATION
OF
THE
ASYMPTOTICS
OF
SOLUTION
OF
THE
BOUNDARY
VALUE
PROBLEM
.
140
4.4.1
THE
PROBLEM
IN
MATRIX
NOTATION
.
140
4.4.2
AUXILIARY
OPERATORS
AND
THEIR
PROPERTIES
.
141
4.4.3
FORMAL
ASYMPTOTICS
OF
THE
SOLUTION
IN
THE
CASE
OF
UNIQUELY
SOLVABLE
LIMIT
PROBLEMS
.
142
4.4.4
A
PARTICULAR
BASIS
IN
THE
COKERNEL
OF
THE
OPERATOR
M
Q
.
144
4.4.5
FORMAL
SOLUTION
IN
THE
CASE
OF
NON-UNIQUE
SOLVABILITY
OF
THE
LIMIT
PROBLEMS
.
149
4.4.6
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
SINGULARLY
PERTURBED
PROBLEM
.
154
CHAPTER
5
VARIANTS
AND
COROLLARIES
OF
THE
ASYMPTOTIC
THEORY
5.1
ESTIMATES
OF
SOLUTIONS
OF
THE
DIRICHLET
PROBLEM
FOR
THE
HELMHOLTZ
OPERATOR
IN
A
DOMAIN
WITH
BOUNDARY
SMOOTHENED
NEAR
A
CORNER
.
157
5.2
SOBOLEV
BOUNDARY
VALUE
PROBLEMS
.
161
5.3
GENERAL
BOUNDARY
VALUE
PROBLEM
IN
A
DOMAIN
WITH
SMALL
HOLES
.
167
5.4
PROBLEMS
WITH
NON-SMOOTH
AND
PARAMETER
DEPENDENT
DATA
.
173
5.4.1
THE
CASE
OF
A
NON-SMOOTH
DOMAIN
.
173
5.4.2
THE
CASE
OF
PARAMETER
DEPENDENT
AUXILIARY
PROBLEMS
.
175
5.4.3
THE
CASE
OF
A
PARAMETER
INDEPENDENT
DOMAIN
.
177
5.5
NON-LOCAL
PERTURBATION
OF
A
DOMAIN
WITH
CONE
VERTICES
.
182
5.5.1
PERTURBATIONS
OF
A
DOMAIN
WITH
SMOOTH
BOUNDARY
.
182
CONTENTS
IX
5.5.2
REGULAR
PERTURBATION
OF
A
DOMAIN
WITH
A
CORNER
.
184
5.5.3
A
NON-LOCAL
SINGULAR
PERTURBATION
OF
A
PLANAR
DOMAIN
WITH
A
CORNER
.
186
5.6
ASYMPTOTICS
OF
SOLUTIONS
TO
BOUNDARY
VALUE
PROBLEMS
IN
LONG
TUBULAR
DOMAINS
.
189
5.6.1
THE
PROBLEM
.
189
5.6.2
LIMIT
PROBLEMS
.
190
5.6.3
SOLVABILITY
OF
THE
ORIGINAL
PROBLEM
.
192
5.6.4
EXPANSION
OF
THE
RIGHT-HAND
SIDES
AND
THE
SET
OF
EXPONENTS
IN
THE
ASYMPTOTICS
.
193
5.6.5
REDISTRIBUTION
OF
DEFECTS
.
195
5.6.6
COEFFICIENTS
IN
THE
ASYMPTOTIC
SERIES
.
197
5.6.7
ESTIMATE
OF
THE
REMAINDER
TERM
.
198
5.6.8
EXAMPLE
.
200
5.7
ASYMPTOTICS
OF
SOLUTIONS
OF
A
QUASI-LINEAR
EQUATION
IN
A
DOMAIN
WITH
SINGULARLY
PERTURBED
BOUNDARY
.
201
5.7.1
A
THREE-DIMENSIONAL
DOMAIN
WITH
A
SMALL
GAP
.
202
5.7.2
A
PLANAR
DOMAIN
WITH
A
SMALL
GAP
.
207
5.7.3
A
DOMAIN
SMOOTHENED
NEAR
A
CORNER
POINT
.
213
5.8
BENDING
OF
AN
ALMOST
POLYGONAL
PLATE
WITH
FREELY
SUPPORTED
BOUNDARY
.
217
5.8.1
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
WITH
CORNERS
.
219
5.8.2
A
SINGULARLY
PERTURBED
DOMAIN
AND
LIMIT
PROBLEMS
.
220
5.8.3
THE
PRINCIPAL
TERM
IN
THE
ASYMPTOTICS
.
221
5.8.4
THE
PRINCIPAL
TERM
IN
THE
ASYMPTOTICS
(CONTINUED)
.
223
PART
III
ASYMPTOTIC
BEHAVIOUR
OF
FUNCTIONALS
ON
SOLUTIONS
OF
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
PERTURBED
NEAR
ISOLATED
BOUNDARY
SINGULARITIES
CHAPTER
6
ASYMPTOTIC
BEHAVIOUR
OF
INTENSITY
FACTORS
FOR
VERTICES
OF
CORNERS
AND
CONES
COMING
CLOSE
6.1
DIRICHLET
'
S
PROBLEM
FOR
LAPLACE
'
S
OPERATOR
.
228
6.1.1
STATEMENT
OF
THE
PROBLEM
.
228
6.1.2
ASYMPTOTIC
BEHAVIOUR
OF
THE
COEFFICIENT
C+
.
229
6.1.3
JUSTIFICATION
OF
THE
ASYMPTOTIC
FORMULA
FOR
THE
COEFFICIENT
(7+
.
230
6.1.4
THE
CASE
G
0
.
231
6.1.5
THE
TWO-DIMENSIONAL
CASE
.
231
6.2
NEUMANN
'
S
PROBLEM
FOR
LAPLACE
'
S
OPERATOR
.
232
6.2.1
STATEMENT
OF
THE
PROBLEM
.
232
6.2.2
BOUNDARY
VALUE
PROBLEMS
.
232
6.2.3
THE
CASE
OF
DISCONNECTED
BOUNDARY
.
234
6.2.4
THE
CASE
OF
CONNECTED
BOUNDARY
.
235
6.3
INTENSITY
FACTORS
FOR
BENDING
OF
A
THIN
PLATE
WITH
A
CRACK
.
235
6.3.1
STATEMENT
OF
THE
PROBLEM
.
235
6.3.2
CLAMPED
CRACKS
(THE
ASYMPTOTIC
BEHAVIOUR
NEAR
CRACK
TIPS)
.
236
X
CONTENTS
6.3.3
FIXEDLY
CLAMPED
CRACKS
(ASYMPTOTIC
BEHAVIOUR
OF
THE
INTENSITY
FACTORS)
.
237
6.3.4
FREELY
SUPPORTED
CRACKS
.
238
6.3.5
FREE
CRACKS
(THE
ASYMPTOTIC
BEHAVIOUR
OF
SOLUTION
NEAR
CRACK
VERTICES)
.
240
6.3.6
FREE
CRACKS
(THE
ASYMPTOTIC
BEHAVIOUR
OF
INTENSITY
FACTORS)
.
240
6.4
ANTIPLANAR
AND
PLANAR
DEFORMATIONS
OF
DOMAINS
WITH
CRACKS
.
243
6.4.1
TORSION
OF
A
BAR
WITH
A
LONGITUDINAL
CRACK
.
243
6.4.2
THE
TWO-DIMENSIONAL
PROBLEM
OF
THE
ELASTICITY
THEORY
IN
A
DOMAIN
WITH
COLLINEAR
CLOSE
CRACKS
.
245
CHAPTER
7
ASYMPTOTIC
BEHAVIOUR
OF
ENERGY
INTEGRALS
FOR
SMALL
PERTURBATIONS
OF
THE
BOUNDARY
NEAR
CORNERS
AND
ISOLATED
POINTS
7.1
ASYMPTOTIC
BEHAVIOUR
OF
SOLUTIONS
OF
THE
PERTURBED
PROBLEM
.
251
7.1.1
THE
UNPERTURBED
BOUNDARY
VALUE
PROBLEM
.
251
7.1.2
PERTURBED
PROBLEM
.
254
7.1.3
THE
SECOND
LIMIT
PROBLEM
.
254
7.1.4
ASYMPTOTIC
BEHAVIOUR
OF
SOLUTIONS
OF
THE
PERTURBED
PROBLEM
.
256
7.1.5
THE
CASE
OF
RIGHT-HAND
SIDES
LOCALIZED
NEAR
A
POINT
.
259
7.2
ASYMPTOTIC
BEHAVIOUR
OF
A
BILINEAR
FORM
.
261
7.2.1
THE
ASYMPTOTIC
BEHAVIOUR
OF
A
BILINEAR
FORM
(THE
GENERAL
CASE)
.
261
7.2.2
ASYMPTOTIC
BEHAVIOUR
OF
A
BILINEAR
FORM
FOR
RIGHT-HAND
SIDES
LOCALIZED
NEAR
A
POINT
.
264
7.2.3
ASYMPTOTIC
BEHAVIOUR
OF
A
QUADRATIC
FORM
.
266
7.3
ASYMPTOTIC
BEHAVIOUR
OF
A
QUADRATIC
FORM
FOR
PROBLEMS
IN
REGIONS
WITH
SMALL
HOLES
.
267
7.3.1
STATEMENT
OF
THE
PROBLEM
.
267
7.3.2
THE
CASE
OF
UNIQUELY
SOLVABLE
BOUNDARY
PROBLEMS
.
267
7.3.3
THE
CASE
OF
THE
CRITICAL
DIMENSION
.
270
CHAPTER
8
ASYMPTOTIC
BEHAVIOUR
OF
ENERGY
INTEGRALS
FOR
PARTICULAR
PROBLEMS
OF
MATHEMATICAL
PHYSICS
8.1
DIRICHLET
'
S
PROBLEM
FOR
LAPLACE
'
S
OPERATOR
.
277
8.1.1
PERTURBATION
OF
A
DOMAIN
NEAR
A
CORNER
OR
CONIC
POINT
.
277
8.1.2
THE
CASE
OF
RIGHT-HAND
SIDES
DEPENDING
ON
.
280
8.1.3
THE
CASE
OF
RIGHT-HAND
SIDES
DEPENDING
ON
X
AND
.
281
8.1.4
DIRICHLET
'
S
PROBLEM
FOR
LAPLACE
'
S
OPERATOR
IN
A
DOMAIN
WITH
A
SMALL
HOLE
.
282
8.1.5
REFINEMENT
OF
THE
ASYMPTOTIC
BEHAVIOUR
.
284
8.1.6
TWO-DIMENSIONAL
DOMAINS
WITH
A
SMALL
HOLE
.
287
8.1.7
DIRICHLET
'
S
PROBLEM
FOR
LAPLACE
'
S
OPERATOR
IN
DOMAINS
WITH
SEVERAL
SMALL
HOLES
.
288
CONTENTS
XI
8.2
NEUMANN
'
S
PROBLEM
IN
DOMAINS
WITH
ONE
SMALL
HOLE
.
291
8.3
DIRICHLET
'
S
PROBLEM
FOR
THE
BIHARMONIC
EQUATION
IN
A
DOMAIN
WITH
SMALL
HOLES
.
293
8.4
VARIATION
OF
ENERGY
DEPENDING
ON
THE
LENGTH
OF
CRACK
.
296
8.4.1
THE
ANTIPLANAR
DEFORMATION
.
296
8.4.2
A
PROBLEM
IN
THE
TWO-DIMENSIONAL
ELASTICITY
.
299
8.5
REMARKS
ON
THE
BEHAVIOUR
OF
SOLUTIONS
OF
PROBLEMS
IN
THE
TWO-DIMENSIONAL
ELASTICITY
NEAR
CORNER
POINTS
.
302
8.5.1
STATEMENT
OF
PROBLEMS
.
302
8.5.2
THE
ASYMPTOTIC
BEHAVIOUR
OF
SOLUTIONS
OF
THE
ANTIPLANAR
DEFORMATION
PROBLEM
.
302
8.5.3
ASYMPTOTIC
BEHAVIOUR
OF
SOLUTIONS
OF
THE
PLANAR
DEFORMATION
PROBLEM
.
303
8.5.4
BOUNDARY
VALUE
PROBLEMS
IN
UNBOUNDED
DOMAINS
.
306
8.6
DERIVATION
OF
ASYMPTOTIC
FORMULAS
FOR
ENERGY
.
308
8.6.1
STATEMENT
OF
PROBLEMS
.
308
8.6.2
ANTIPLANAR
DEFORMATION
.
309
8.6.3
PLANAR
DEFORMATION
.
310
8.6.4
REFINEMENT
OF
THE
ASYMPTOTIC
FORMULA
FOR
ENERGY
.
311
8.6.5
DEFECT
IN
THE
MATERIAL
NEAR
VERTEX
OF
THE
CRACK
.
313
PART
IV
ASYMPTOTIC
BEHAVIOUR
OF
EIGENVALUES
OF
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
WITH
SMALL
HOLES
CHAPTER
9
ASYMPTOTIC
EXPANSIONS
OF
EIGENVALUES
OF
CLASSIC
BOUNDARY
VALUE
PROBLEMS
9.1
ASYMPTOTIC
BEHAVIOUR
OF
THE
FIRST
EIGENVALUE
OF
A
MIXED
BOUNDARY
VALUE
PROBLEM
.
318
9.1.1
STATEMENT
OF
THE
PROBLEM
.
318
9.1.2
THE
THREE-DIMENSIONAL
CASE
(FORMAL
ASYMPTOTIC
REPRESENTATION)
.
319
9.1.3
THE
PLANAR
CASE
(FORMAL
ASYMPTOTIC
REPRESENTATION)
.
322
9.1.4
JUSTIFICATION
OF
ASYMPTOTIC
EXPANSIONS
IN
THE
THREE-DIMENSIONAL
CASE
.
326
9.1.5
JUSTIFICATION
OF
ASYMPTOTIC
EXPANSIONS
IN
THE
TWO-DIMENSIONAL
CASE
.
329
9.2
ASYMPTOTIC
EXPANSIONS
OF
EIGENVALUES
OF
OTHER
BOUNDARY
VALUE
PROBLEMS
.
331
9.2.1
DIRICHLET
'
S
PROBLEM
IN
A
THREE-DIMENSIONAL
DOMAIN
WITH
A
SMALL
HOLE
.
331
9.2.2
MIXED
BOUNDARY
VALUE
PROBLEM
IN
DOMAINS
WITH
SEVERAL
SMALL
HOLES
.
334
9.2.3
MIXED
BOUNDARY
VALUE
PROBLEM
WITH
NEUMANN
'
S
CONDITION
ON
THE
BOUNDARY
OF
SMALL
HOLE
.
337
9.2.4
DIRICHLET
'
S
PROBLEM
ON
A
RIEMANNIAN
MANIFOLD
WITH
A
SMALL
HOLE
.
340
XII
CONTENTS
9.3
ASYMPTOTIC
REPRESENTATIONS
OF
EIGENVALUES
OF
PROBLEMS
OF
THE
ELASTICITY
THEORY
FOR
BODIES
WITH
SMALL
INCLUSIONS
AND
HOLES
.
342
9.3.1
STATEMENT
OF
THE
PROBLEM
.
342
9.3.2
STRUCTURE
OF
THE
ASYMPTOTIC
REPRESENTATION
.
343
9.3.3
PARTICULAR
SOLUTIONS
OF
THE
BOUNDARY
LAYER
PROBLEM
.
343
9.3.4
PERTURBATION
OF
THE
EIGENVALUE
AO
.
347
9.3.5
PROBLEM
IN
THE
TWO-DIMENSIONAL
ELASTICITY
(ONE
HOLE
WITH
A
FREE
SURFACE)
.
349
CHAPTER
10
HOMOGENEOUS
SOLUTIONS
OF
BOUNDARY
VALUE
PROBLEMS
IN
THE
EXTERIOR
OF
A
THIN
CONE
10.1
FORMAL
ASYMPTOTIC
REPRESENTATION
.
355
10.1.1
STATEMENT
OF
THE
PROBLEM
.
355
10.1.2
THE
CASE
N
-
1
2M
.
356
10.1.3
THE
CASE
N
-
1
=
2M
.
360
10.2
INVERSION
OF
THE
PRINCIPAL
PART
OF
AN
OPERATOR
PENCIL
ON
THE
UNIT
SPHERE
WITH
A
SMALL
HOLE.
AN
AUXILIARY
PROBLEM
WITH
MATRIX
OPERATOR
.
363
10.2.1
"
NEARLY
INVERSE
"
OPERATOR
(THE
CASE
2M
N
-
1)
.
363
10.2.2
"
NEARLY
INVERSE
"
OPERATOR
(THE
CASE
2M
=
N
-
1)
.
367
10.2.3
REDUCTION
TO
A
PROBLEM
WITH
A
MATRIX
OPERATOR
(THE
CASE
2M
N
-
1)
.
371
10.2.4
REDUCTION
TO
A
PROBLEM
WITH
A
MATRIX
OPERATOR
(THE
CASE
2M
=
N
-
1)
.
373
10.3
JUSTIFICATION
OF
THE
ASYMPTOTIC
BEHAVIOUR
OF
EIGENVALUES
(THE
CASE
2M
N
-
1)
.
374
10.4
JUSTIFICATION
OF
THE
ASYMPTOTIC
BEHAVIOUR
OF
EIGENVALUES
(THE
CASE
2M
=
N
-
1)
.
379
10.5
EXAMPLES
AND
COROLLARIES
.
388
10.5.1
A
SCALAR
OPERATOR
.
388
10.5.2
LAME
'
S
AND
STOKES
'
SYSTEMS
.
389
10.5.3
CONTINUITY
AT
THE
CONE
VERTEX
OF
SOLUTION
OF
DIRICHLET
'
S
PROBLEM
.
390
10.6
EXAMPLES
OF
DISCONTINUOUS
SOLUTIONS
TO
DIRICHLET
'
S
PROBLEM
IN
DOMAINS
WITH
A
CONIC
POINT
.
391
10.6.1
EQUATION
OF
SECOND
ORDER
WITH
DISCONTINUOUS
SOLUTIONS
.
391
10.6.2
DIRICHLET
'
S
PROBLEM
FOR
AN
ELLIPTIC
EQUATION
OF
THE
FOURTH
ORDER
WITH
REAL
COEFFICIENTS
.
393
10.7
SINGULARITIES
OF
SOLUTIONS
OF
NEUMANN
'
S
PROBLEM
.
395
10.7.1
INTRODUCTION
.
395
10.7.2
FORMAL
ASYMPTOTIC
REPRESENTATION
.
396
10.8
JUSTIFICATION
OF
THE
ASYMPTOTIC
FORMULAS
.
400
10.8.1
MULTIPLICITY
OF
THE
SPECTRUM
NEAR
THE
POINT
A
=
2
.
400
10.8.2
NEARLY
INVERSE
OPERATOR
FOR
NEUMANN
'
S
PROBLEM
IN
G
E
.
401
10.8.3
JUSTIFICATION
OF
ASYMPTOTIC
REPRESENTATION
OF
EIGENVALUES
.
406
CONTENTS
XIII
COMMENTS
ON
PARTS
I-IV
COMMENTS
ON
PART
I
.
411
CHAPTER
1
.
411
CHAPTER
2
.
411
COMMENTS
ON
PART
II
.
411
CHAPTER
3
.
411
CHAPTER
4
.
412
CHAPTER
5
.
412
COMMENTS
ON
PART
III
.
412
CHAPTER
6
.
412
CHAPTER
7
.
408
CHAPTER
8
.
412
COMMENTS
ON
PART
IV
.
412
CHAPTER
9
.
412
CHAPTER
10
.
412
LIST
OF
SYMBOLS
.
413
1.
BASIC
SYMBOLS
.
413
2.
SYMBOLS
FOR
FUNCTION
SPACES
AND
RELATED
CONCEPTS
.
414
3.
SYMBOLS
FOR
FUNCTIONS,
DISTRIBUTIONS
AND
RELATED
CONCEPTS
.
415
4.
OTHER
SYMBOLS
.
415
REFERENCES
.
417
INDEX
.
433
XIV
CONTENTS
VOLUME
II
PREFACE
.
XXI
PART
V
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
PERTURBED
NEAR
MULTIDIMENSIONAL
SINGULARITIES
OF
THE
BOUNDARY
CHAPTER
11
BOUNDARY
VALUE
PROBLEMS
IN
DOMAINS
WITH
EDGES
ON
THE
BOUNDARY
11.1
THE
DIRICHLET
PROBLEM
FOR
THE
LAPLACE
OPERATOR
.
3
11.1.1
STATEMENT
OF
THE
PROBLEM
.
3
11.1.2
MODEL
PROBLEM
IN
A
WEDGE
.
4
11.1.3
BOUNDARY
VALUE
PROBLEM
IN
Q
.
8
11.2
THE
NEUMANN
PROBLEM
FOR
THE
LAPLACE
OPERATOR
.
9
11.2.1
STATEMENT
OF
THE
PROBLEM
.
9
11.2.2
MODEL
PROBLEMS
.
10
11.2.3
SOLVABILITY
OF
THE
NEUMANN
PROBLEM
.
14
11.2.4
THE
PROBLEM
IN
A
DOMAIN
WITH
A
CONTOUR
EXCLUDED
.
15
11.3
THE
ASYMPTOTICS
NEAR
AN
EDGE
OF
SOLUTIONS
TO
BOUNDARY
VALUE
PROBLEMS
FOR
THE
LAPLACE
OPERATOR
.
15
11.3.1
ESTIMATES
FOR
THE
DERIVATIVES
OF
SOLUTIONS
ALONG
AN
EDGE
.
15
11.3.2
THE
ASYMPTOTICS
OF
SOLUTIONS
TO
THE
DIRICHLET
PROBLEM
.
16
11.3.3
THE
ASYMPTOTICS
OF
SOLUTIONS
TO
THE
NEUMANN
PROBLEM
.
18
11.3.4
THE
ASYMPTOTICS
OF
SOLUTIONS
TO
THE
BOUNDARY
VALUE
PROBLEM
IN
A
DOMAIN
WITH
EXCLUDED
CONTOUR
.
18
11.4
SOBOLEV
PROBLEMS
.
19
11.4.1
STATEMENT
OF
THE
PROBLEM
.
19
11.4.2
SOLVABILITY
OF
THE
PROBLEM
.
19
11.4.3
ASYMPTOTICS
OF
A
SOLUTION
.
20
CHAPTER
12
ASYMPTOTICS
OF
SOLUTIONS
TO
CLASSICAL
BOUNDARY
VALUE
PROBLEMS
IN
A
DOMAIN
WITH
THIN
CAVITIES
12.1
THE
ASYMPTOTICS
OF
A
SOLUTION
OF
THE
NEUMANN
PROBLEM
IN
THE
EXTERIOR
OF
A
THIN
TUBE
.
23
12.1.1
STATEMENT
OF
THE
PROBLEM
.
23
12.1.2
PRINCIPAL
TERMS
OF
ASYMPTOTICS
.
24
12.1.3
ASYMPTOTIC
SERIES
.
30
12.2
THE
ASYMPTOTICS
OF
SOLUTIONS
TO
THE
DIRICHLET
PROBLEM
IN
THE
EXTERIOR
OF
A
THIN
TUBE
.
35
12.2.1
STATEMENT
OF
THE
PROBLEM
AND
THE
BOUNDARY
LAYER
.
35
12.2.2
THE
INTEGRAL
EQUATION
ON
AL
.
36
12.2.3
JUSTIFICATION
OF
THE
ASYMPTOTICS
.
39
12.2.4
ASYMPTOTICS
OF
THE
CAPACITY
OF
A
THIN
TOROIDAL
DOMAIN
.
41
CONTENTS
XV
12.3
STRESS
AND
STRAIN
STATE
OF
THE
SPACE
WITH
A
THIN
TOROIDAL
INCLUSION
.
42
12.3.1
STATEMENT
OF
THE
PROBLEM
.
42
12.3.2
PRELIMINARIES
.
42
12.3.3
ASYMPTOTICS
OF
THE
SOLUTION
.
45
12.4
ASYMPTOTICS
OF
SOLUTIONS
TO
THE
DIRICHLET
PROBLEM
IN
A
PLANE
DOMAIN
WITH
A
THIN
CAVITY
.
48
12.4.1
STATEMENT
OF
THE
PROBLEM
.
48
12.4.2
A
CAVITY
WHOSE
SHORES
ENCOUNTER
EACH
OTHER
AT
ZERO
ANGLES
.
48
12.4.3
A
THIN
CAVITY
WITH
SMOOTH
BOUNDARY
.
50
12.4.4
A
REMARK
ON
APPLICATION
OF
THE
RESULTS
OF
CHAPTER
4
.
54
12.5
ASYMPTOTICS
OF
SOLUTIONS
TO
THE
DIRICHLET
PROBLEM
IN
A
THREE-DIMENSIONAL
DOMAIN
WITH
A
THIN
CAVITY
.
55
12.5.1
STATEMENT
OF
THE
PROBLEM
.
56
12.5.2
ASYMPTOTICS
OF
THE
SOLUTION
IN
THE
EXTERIOR
OF
Q
E
.
56
12.5.3
THE
TWO-DIMENSIONAL
BOUNDARY
LAYER
.
57
12.5.4
CONSTRUCTION
OF
THE
FUNCTION
7
.
58
12.5.5
BOUNDARY
LAYER
NEAR
THE
ENDPOINTS
OF
SEGMENT
M
(THE
FIRST
TERM)
.
61
12.5.6
BOUNDARY
LAYER
NEAR
THE
ENDPOINTS
OF
THE
SEGMENT
M
(THE
SECOND
TERM)
.
63
12.5.7
JUSTIFICATION
OF
THE
ASYMPTOTICS
.
65
12.5.8
ASYMPTOTICS
OF
CAPACITY
OF
THE
"
ELLIPSOID
"
Q
E
.
68
12.5.9
THE
CASE
OF
A
ROTATION
ELLIPSOID
.
69
12.6
SMOOTHING
THE
BOUNDARY
NEAR
AN
EDGE
.
70
12.6.1
THE
DOMAIN
.
71
12.6.2
THE
PRINCIPAL
TERM
OF
ASYMPTOTICS
.
72
12.6.3
THE
COMPLETE
ASYMPTOTIC
EXPANSION
.
73
CHAPTER
13
ASYMPTOTICS
OF
SOLUTIONS
TO
THE
DIRICHLET
PROBLEM
FOR
HIGH
ORDER
EQUATIONS
IN
A
DOMAIN
WITH
A
THIN
TUBE
EXCLUDED
13.1
STATEMENT
OF
THE
PROBLEM
.
76
13.2
THE
CASE
OF
NONCRITICAL
DIMENSION
.
77
13.3
THE
CASE
OF
CRITICAL
DIMENSION
(EXPANSION
IN
(LOGS)
-1
)
.
81
13.4
THE
CASE
OF
CRITICAL
DIMENSION
(EXPANSION
IN
POWERS
OF
E)
.
88
13.4.1
STRUCTURE
OF
THE
ASYMPTOTIC
EXPANSION
.
88
13.4.2
STRUCTURE
OF
THE
FORMAL
SERIES
(1)
.
89
13.4.3
ASYMPTOTIC
INVERSION
OF
THE
OPERATOR
AI
LOG
S
+
A2
+
A
.
91
13.4.4
THE
POWER
ASYMPTOTIC
SERIES
FOR
THE
SOLUTION
.
95
13.4.5
MORE
ON
THE
DIRICHLET
PROBLEM
FOR
THE
LAPLACE
OPERATOR
IN
THE
EXTERIOR
OF
A
TUBE
.
98
XVI
CONTENTS
PART
VI
BEHAVIOUR
OF
SOLUTIONS
OF
BOUNDARY
VALUE
PROBLEMS
IN
THIN
DOMAINS
CHAPTER
14
THE
DIRICHLET
PROBLEM
IN
DOMAINS
WITH
THIN
LIGAMENTS
14.1
THE
PRINCIPAL
TERM
IN
THE
ASYMPTOTICS
OF
SOLUTION
.
105
14.1.1
STATEMENT
OF
THE
PROBLEM
(THE
CASE
OF
TWO
POINTS
ON
SMOOTH
SURFACES
APPROACHING
EACH
OTHER)
.
105
14.1.2
THE
LIMIT
BOUNDARY
VALUE
PROBLEM
.
105
14.1.3
THE
ASYMPTOTICS
OF
SOLUTIONS
TO
THE
ORIGINAL
PROBLEM
.
109
14.2
COMPLETE
ASYMPTOTIC
EXPANSIONS
OF
SOLUTIONS
.
ILL
14.2.1
THE
CASE
OF
A
QUASICYLINDRICAL
DOMAIN
.
ILL
14.2.2
ASYMPTOTICS
OF
SOLUTIONS
TO
THE
DIRICHLET
PROBLEM
.
114
14.3
ASYMPTOTICS
OF
SOLUTIONS
FOR
NONSMOOTH
RIGHT-HAND
SIDE
TERMS
.
114
14.3.1
THE
SECOND
LIMIT
PROBLEM
.
114
14.3.2
ASYMPTOTICS
OF
SOLUTIONS
.
116
14.4
LIGAMENTS
OF
A
DIFFERENT
FORM
.
120
14.4.1
TWO
CLOSE
CONICAL
POINTS
.
120
14.4.2
COMPONENTS
OF
THE
BOUNDARY
ADHERING
AT
A
LARGE
SET
.
122
14.5
ASYMPTOTICS
OF
THE
CONDENSER
CAPACITY
.
124
14.5.1
COMPONENTS
OF
THE
BOUNDARY
ADHERING
AT
ONE
POINT
.
124
14.5.2
OTHER
CONDENSERS
.
126
14.5.3
THE
COMPLETE
ASYMPTOTICS
OF
CAPACITY
.
128
CHAPTER
15
BOUNDARY
VALUE
PROBLEMS
OF
MATHEMATICAL
PHYSICS
IN
THIN
DOMAINS
15.1
BOUNDARY
VALUE
PROBLEMS
FOR
THE
LAPLACE
OPERATOR
IN
A
THIN
RECTANGLE
.
131
15.1.1
STATEMENTS
OF
THE
PROBLEMS
.
131
15.1.2
ASYMPTOTICS
OF
SOLUTION
TO
THE
DIRICHLET
PROBLEM
.
132
15.1.3
ASYMPTOTICS
OF
SOLUTION
TO
THE
MIXED
PROBLEM
.
133
15.2
THE
PRINCIPAL
TERM
IN
ASYMPTOTICS
OF
THE
SOLUTION
TO
A
BOUNDARY
VALUE
PROBLEM
FOR
A
SYSTEM
OF
SECOND
ORDER
EQUATIONS
IN
A
CYLINDER
OF
SMALL
HEIGHT
.
136
15.2.1
STATEMENT
OF
THE
PROBLEM
.
136
15.2.2
AUXILIARY
CONSTRUCTIONS
.
138
15.2.3
ASYMPTOTICS
OF
THE
SOLUTION
.
140
15.2.4
PROPERTIES
OF
THE
LIMIT
OPERATOR
.
142
15.2.5
UNIQUE
SOLVABILITY
OF
THE
LIMIT
PROBLEM
.
143
15.2.6
JUSTIFICATION
OF
THE
ASYMPTOTIC
EXPANSION
OF
THE
SOLUTION
.
145
15.3
APPLICATIONS
OF
THEOREM
15.2.9
TO
PARTICULAR
BOUNDARY
VALUE
PROBLEMS
.
149
15.4
ANTIPLANAR
SHEAR
AND
FLOW
OF
AN
IDEAL
FLUID
IN
A
THIN
DOMAIN
WITH
A
LONGITUDINAL
CUT
.
153
15.4.1
STATEMENT
OF
THE
PROBLEM
.
153
15.4.2
THE
TWO-DIMENSIONAL
CASE
.
154
15.4.3
BOUNDARY
LAYERS
.
155
CONTENTS
XVII
15.4.4
A
SUPPLEMENTARY
LIMIT
PROBLEM
AND
ASYMPTOTICS
OF
THE
INTENSITY
FACTOR
.
157
15.4.5
THE
THREE-DIMENSIONAL
CASE
.
158
15.4.6
EXAMPLES
.
159
15.5
INTENSITY
FACTORS
FOR
CLOSE
PARALLEL
CRACKS
.
161
15.5.1
STATEMENT
OF
THE
PROBLEM
.
161
15.5.2
ASYMPTOTICS
OF
THE
SOLUTION
INSIDE
AND
OUTSIDE
THE
STRIP
BETWEEN
CRACKS
.
162
15.5.3
BOUNDARY
LAYERS
NEAR
TIPS
OF
THE
CUT
.
163
15.5.4
ESTIMATE
ON
THE
REMAINDER
IN
ASYMPTOTICS
.
166
15.5.5
ASYMPTOTICS
OF
THE
INTENSITY
FACTORS
.
166
15.5.6
SHIFTED
CRACKS
.
168
CHAPTER
16
GENERAL
ELLIPTIC
PROBLEMS
IN
THIN
DOMAINS
16.1
LIMIT
PROBLEMS
.
171
16.1.1
STATEMENT
OF
THE
PROBLEM
.
171
16.1.2
STRUCTURE
OF
DIFFERENTIAL
OPERATORS
.
172
16.1.3
THE
ELLIPTICITY
CONDITION
.
173
16.1.4
THE
FIRST
LIMIT
PROBLEM
.
174
16.1.5
THE
SECOND
LIMIT
PROBLEM
.
175
16.1.6
THE
THIRD
LIMIT
PROBLEM
.
177
16.2
ASYMPTOTICS
OF
SOLUTIONS
.
180
16.2.1
THE
FREDHOLM
PROPERTY
OF
THE
ORIGINAL
PROBLEM
.
180
16.2.2
THE
CASE
WHEN
LIMIT
PROBLEMS
ARE
UNIQUELY
SOLVABLE
.
182
16.2.3
SOLUTIONS
TO
THE
THIRD
LIMIT
PROBLEM
.
185
16.2.4
ASYMPTOTICS
IN
THE
CASE
WHEN
K
+
K
K
0
.
188
16.3
EXAMPLES
.
192
16.4
BENDING
OF
A
THIN
PLATE
.
197
16.4.1
STATEMENT
OF
THE
PROBLEM
.
197
16.4.2
THE
FIRST
TWO
LIMIT
PROBLEMS
.
197
16.4.3
THE
COMPLEMENTARY
LIMIT
PROBLEM
.
198
16.4.4
THE
BOUNDARY
LAYER
.
199
16.4.5
BOUNDARY
CONDITIONS
IN
THE
THIRD
LIMIT
PROBLEM
.
206
16.4.6
ASYMPTOTICS
OF
THE
SOLUTION
.
206
PART
VII
ELLIPTIC
BOUNDARY
VALUE
PROBLEMS
WITH
OSCILLATING
COEFFICIENTS
OR
BOUNDARY
OF
DOMAIN
CHAPTER
17
ELLIPTIC
BOUNDARY
VALUE
PROBLEMS
WITH
RAPIDLY
OSCILLATING
COEFFICIENTS
17.1
HOMOGENIZATION
OF
DIFFERENTIAL
EQUATION
.
211
17.1.1
STATEMENT
OF
THE
PROBLEM
.
211
17.1.2
THE
LIMIT
PROBLEM
IN
THE
CELL
.
212
17.1.3
THE
HOMOGENIZED
EQUATION
.
213
17.1.4
ASYMPTOTIC
SERIES
.
215
XVIII
CONTENTS
17.2
BOUNDARY
LAYER
FOR
THE
DIRICHLET
PROBLEM
.
216
17.2.1
THE
BOUNDARY
VALUE
PROBLEM
FOR
THE
BOUNDARY
LAYER
.
216
17.2.2
CONDITIONS
ON
THE
BOUNDARY
LAYER
.
218
17.3
BOUNDARY
LAYER
FOR
THE
NEUMANN
PROBLEM
.
222
17.4
JUSTIFICATION
OF
ASYMPTOTIC
EXPANSIONS
.
226
17.5
ELLIPTIC
BOUNDARY
VALUE
PROBLEMS
WITH
PERIODIC
COEFFICIENTS
IN
A
CYLINDER
.
229
17.5.1
MODEL
PROBLEM
IN
A
CYLINDER
.
229
17.5.2
PROBLEM
WITH
COMPLEX
PARAMETER
.
230
17.5.3
ANALOG
OF
THE
FOURIER
TRANSFORM
.
231
17.5.4
UNIQUE
SOLVABILITY
OF
THE
MODEL
PROBLEM
.
233
17.5.5
ASYMPTOTICS
OF
SOLUTIONS
.
234
CHAPTER
18
PARADOXES
OF
LIMIT
PASSAGE
IN
SOLUTIONS
OF
BOUNDARY
VALUE
PROBLEMS
WHEN
SMOOTH
DOMAINS
ARE
APPROXIMATED
BY
POLYGONS
18.1
APPROXIMATION
TO
A
FREELY
SUPPORTED
CONVEX
PLATE
.
238
18.1.1
STATEMENT
OF
THE
PROBLEM
AND
DESCRIPTION
OF
THE
RESULTS
.
238
18.1.2
FORMAL
ASYMPTOTICS
.
239
18.1.3
JUSTIFICATION
OF
THE
ASYMPTOTICS
.
243
18.1.4
CONCENTRATED
MOMENTS
AT
THE
VERTICES
OF
THE
POLYGON
.
247
18.2
APPROXIMATION
OF
A
HOLE
IN
A
FREELY
SUPPORTED
PLATE
.
249
18.2.1
STATEMENT
OF
THE
PROBLEM
.
249
18.2.2
ASYMPTOTICS
OF
THE
SOLUTION
FOR
THE
PLATE
WITH
A
POLYGONAL
HOLE
.
250
18.2.3
POINT
MOMENTS
AT
THE
VERTICES
OF
THE
POLYGON
.
252
18.3
PASSAGE
TO
CONDITIONS
OF
RIGID
SUPPORT
.
254
CHAPTER
19
HOMOGENIZATION
OF
A
DIFFERENTIAL
OPERATOR
ON
A
FINE
PERIODIC
NET
OF
CURVES
19.1
STATEMENT
OF
THE
PROBLEM
ON
A
NET
.
259
19.1.1
THE
NET
S
.
259
19.1.2
THE
NET
S
E
.
260
19.2
THE
PRINCIPAL
TERM
OF
ASYMPTOTICS
.
261
19.2.1
THE
FORMAL
ASYMPTOTICS
.
261
19.2.2
JUSTIFICATION
OF
THE
ASYMPTOTICS
.
262
19.2.3
ASYMPTOTICS
OF
SOLUTIONS
TO
NONSTATIONARY
PROBLEMS
.
266
19.3
COMPUTATION
OF
COEFFICIENTS
OF
THE
HOMOGENIZED
OPERATOR
AND
THEIR
PROPERTIES
.
267
19.3.1
DEFINITION
OF
COEFFICIENTS
OF
THE
HOMOGENIZED
OPERATOR
.
267
19.3.2
ELLIPTICITY
OF
THE
HOMOGENIZED
OPERATOR
.
269
19.3.3
EXAMPLES
OF
HOMOGENIZED
OPERATORS
.
270
19.4
THE
COMPLETE
ASYMPTOTIC
EXPANSION
.
273
19.4.1
ASYMPTOTIC
SOLUTION
OUTSIDE
A
NEIGHBORHOOD
OF
THE
BOUNDARY
.
273
19.4.2
CONSTRUCTION
OF
THE
BOUNDARY
LAYER
.
274
19.4.3
DEFINITION
OF
CONSTANTS
IN
THE
BOUNDARY
LAYER
.
276
CONTENTS
XIX
19.5
ASYMPTOTICS
OF
THE
SOLUTION
TO
A
BOUNDARY
VALUE
PROBLEM
ON
A
NET
LOCATED
IN
A
CYLINDER
.
278
19.5.1
THE
BOUNDARY
VALUE
PROBLEM
IN
A
CYLINDER
.
278
19.5.2
THE
BOUNDARY
VALUE
PROBLEM
ON
A
CELL
.
279
19.5.3
ASYMPTOTIC
EXPANSION
OF
SOLUTIONS
TO
THE
PROBLEM
IN
A
CYLINDER
.
280
CHAPTER
20
HOMOGENIZATION
OF
EQUATIONS
ON
A
FINE
PERIODIC
GRID
20.1
HOMOGENIZATION
OF
DIFFERENCE
EQUATIONS
.
283
20.1.1
A
GRID
IN
AND
THE
INTERACTION
SET
OF
ITS
POINTS
.
283
20.1.2
STATEMENT
OF
THE
PROBLEM
.
284
20.1.3
SOLVABILITY
OF
THE
BOUNDARY
VALUE
PROBLEM
.
285
20.1.4
THE
LEADING
TERMS
IN
ASYMPTOTICS
.
286
20.1.5
ASYMPTOTICS
OF
THE
SOLUTION
TO
THE
NONSTATIONARY
PROBLEM
.
288
20.2
CALCULATION
OF
COEFFICIENTS
OF
THE
HOMOGENIZED
OPERATOR
AND
THEIR
PROPERTIES
.
288
20.3
CRYSTALLINE
GRID
.
291
20.3.1
EQUATIONS
OF
THE
ELASTICITY
THEORY
.
291
20.3.2
EXAMPLES
OF
HOMOGENIZED
OPERATORS
.
293
COMMENTS
ON
PARTS
V-VII
COMMENTS
ON
PART
V
.
297
CHAPTER
11
.
297
CHAPTER
12
.
297
CHAPTER
13
.
297
COMMENTS
ON
PART
VI
.
298
CHAPTER
14
.
298
CHAPTER
15
.
298
CHAPTER
16
.
298
COMMENTS
ON
PART
VII
.
298
CHAPTER
17
.
298
CHAPTER
18
.
299
CHAPTER
19
.
299
CHAPTER
20
.
299
LIST
OF
SYMBOLS
.
301
REFERENCES
.
305
INDEX
.
321 |
| any_adam_object | 1 |
| author | Mazʹja, Vladimir Gilelevič 1937- Nazarov, Sergej A. Plamenevskij, Boris A. |
| author_GND | (DE-588)121490602 |
| author_facet | Mazʹja, Vladimir Gilelevič 1937- Nazarov, Sergej A. Plamenevskij, Boris A. |
| author_role | aut aut aut |
| author_sort | Mazʹja, Vladimir Gilelevič 1937- |
| author_variant | v g m vg vgm s a n sa san b a p ba bap |
| building | Verbundindex |
| bvnumber | BV013152948 |
| classification_rvk | SK 620 |
| ctrlnum | (OCoLC)313838248 (DE-599)BVBBV013152948 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV013152948 |
| illustrated | Not Illustrated |
| indexdate | 2025-04-15T08:01:01Z |
| institution | BVB |
| isbn | 3764363983 |
| language | English German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008961722 |
| oclc_num | 313838248 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-29T DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-29T DE-11 |
| physical | XXIII, 323 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Operator theory |
| series2 | Operator theory |
| spelling | Mazʹja, Vladimir Gilelevič 1937- Verfasser (DE-588)121490602 aut Asymptotic theory of elliptic boundary value problems in singularly perturbed domains 2 Vladimir Maz'ya ; Serguei Nazarov ; Boris Plamenevskij. Transl. from the German by Georg Heinig and Christian Posthoff (2000) Basel [u.a.] Birkhäuser XXIII, 323 S. txt rdacontent n rdamedia nc rdacarrier Operator theory 112 Nazarov, Sergej A. Verfasser aut Plamenevskij, Boris A. Verfasser aut (DE-604)BV013152946 2 Operator theory 112 (DE-604)BV000000970 112 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008961722&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mazʹja, Vladimir Gilelevič 1937- Nazarov, Sergej A. Plamenevskij, Boris A. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains Operator theory |
| title | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains |
| title_alt | Asimptotika rešenij elliptičeskich kraevych zadač pri singuljarnych vozmuščenijach oblasti |
| title_auth | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains |
| title_exact_search | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains |
| title_full | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains 2 Vladimir Maz'ya ; Serguei Nazarov ; Boris Plamenevskij. Transl. from the German by Georg Heinig and Christian Posthoff |
| title_fullStr | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains 2 Vladimir Maz'ya ; Serguei Nazarov ; Boris Plamenevskij. Transl. from the German by Georg Heinig and Christian Posthoff |
| title_full_unstemmed | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains 2 Vladimir Maz'ya ; Serguei Nazarov ; Boris Plamenevskij. Transl. from the German by Georg Heinig and Christian Posthoff |
| title_short | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains |
| title_sort | asymptotic theory of elliptic boundary value problems in singularly perturbed domains |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008961722&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013152946 (DE-604)BV000000970 |
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