Foundations of the classical theory of partial differential equations:
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| Previous Title: | Encyclopaedia of mathematical sciences ; 30 |
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| Main Authors: | , |
| Format: | Book |
| Language: | English |
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Berlin [u.a.]
Springer
1998
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| Edition: | 1. ed., 2. printing |
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| Item Description: | Erstdruck u.d.T.: Partial differential equations ; 1 |
| Physical Description: | 259 S. |
| ISBN: | 3540638253 |
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| 035 | |a (OCoLC)478751152 | ||
| 035 | |a (DE-599)BVBBV011791582 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 044 | |a gw |c DE | ||
| 049 | |a DE-703 |a DE-824 |a DE-384 |a DE-739 |a DE-91G |a DE-83 |a DE-188 | ||
| 084 | |a SK 540 |0 (DE-625)143245: |2 rvk | ||
| 084 | |a MAT 350f |2 stub | ||
| 084 | |a 35-01 |2 msc | ||
| 100 | 1 | |a Egorov, Jurij V. |d 1938- |e Verfasser |0 (DE-588)121177181 |4 aut | |
| 245 | 1 | 0 | |a Foundations of the classical theory of partial differential equations |c Yu. V. Egorov ; M. A. Shubin |
| 250 | |a 1. ed., 2. printing | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1998 | |
| 300 | |a 259 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Erstdruck u.d.T.: Partial differential equations ; 1 | ||
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Šubin, Michail A. |d 1944- |e Verfasser |0 (DE-588)121177211 |4 aut | |
| 780 | 0 | 0 | |i 1. Aufl. u.d.T. |t Encyclopaedia of mathematical sciences ; 30 |
| 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=007960363&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-007960363 | |
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| adam_text |
CONTENTS
PREFACE
.
6
CHAPTER
1.
BASIC
CONCEPTS
.
7
§1.
BASIC
DEFINITIONS
AND
EXAMPLES
.
7
1.1.
THE
DEFINITION
OF
A
LINEAR
PARTIAL
DIFFERENTIAL
EQUATION
.
7
1.2.
THE
ROLE
OF
PARTIAL
DIFFERENTIAL
EQUATIONS
IN
THE
MATHEMATICAL
MODELING
OF
PHYSICAL
PROCESSES
.
7
1.3.
DERIVATION
OF
THE
EQUATION
FOR
THE
LONGITUDINAL
ELASTIC
VIBRATIONS
OF
A
ROD
.
8
1.4.
DERIVATION
OF
THE
EQUATION
OF
HEAT
CONDUCTION
.
9
1.5.
THE
LIMITS
OF
APPLICABILITY
OF
MATHEMATICAL
MODELS
.
10
1.6.
INITIAL
AND
BOUNDARY
CONDITIONS
.
11
1.7.
EXAMPLES
OF
LINEAR
PARTIAL
DIFFERENTIAL
EQUATIONS
.
12
1.8.
THE
CONCEPT
OF
WELL-POSEDNESS
OF
A
BOUNDARY-VALUE
PROBLEM.
THE
CAUCHY
PROBLEM
.
21
§2.
THE
CAUCHY-KOVALEVSKAYA
THEOREM
AND
ITS
GENERALIZATIONS
.
28
2.1.
THE
CAUCHY-KOVALEVSKAYA
THEOREM
.
28
2.2.
AN
EXAMPLE
OF
NONEXISTENCE
OF
AN
ANALYTIC
SOLUTION
.
31
2.3.
SOME
GENERALIZATIONS
OF
THE
CAUCHY-KOVALEVSKAYA
THEOREM.
CHARACTERISTICS
.
31
2.4.
OVSYANNIKOV
'
S
THEOREM
.
33
2.5.
HOLMGREN
'
S
THEOREM
.
35
2
CONTENTS
§3.
CLASSIFICATION
OF
LINEAR
DIFFERENTIAL
EQUATIONS.
REDUCTION
TO
CANONICAL
FORM
AND
CHARACTERISTICS
.
37
3.1.
CLASSIFICATION
OF
SECOND-ORDER
EQUATIONS
AND
THEIR
REDUCTION
TO
CANONICAL
FORM
AT
A
POINT
.
37
3.2.
CHARACTERISTICS
OF
SECOND-ORDER
EQUATIONS
AND
REDUCTION
TO
CANONICAL
FORM
OF
SECOND-ORDER
EQUATIONS
WITH
TWO
INDEPENDENT
VARIABLES
.
39
3.3.
ELLIPTICITY,
HYPERBOLICITY,
AND
PARABOLICITY
FOR
GENERAL
LINEAR
DIFFERENTIAL
EQUATIONS
AND
SYSTEMS
.
41
3.4.
CHARACTERISTICS
AS
SOLUTIONS
OF
THE
HAMILTON-JACOBI
EQUATION
.
45
CHAPTER
2.
THE
CLASSICAL
THEORY
.
47
§1.
DISTRIBUTIONS
AND
EQUATIONS
WITH
CONSTANT
COEFFICIENTS
.
47
1.1.
THE
CONCEPT
OF
A
DISTRIBUTION
.
47
1.2.
THE
SPACES
OF
TEST
FUNCTIONS
AND
DISTRIBUTIONS
.
48
1.3.
THE
TOPOLOGY
IN
THE
SPACE
OF
DISTRIBUTIONS
.
51
1.4.
THE
SUPPORT
OF
A
DISTRIBUTION.
THE
GENERAL
FORM
OF
DISTRIBUTIONS
.
53
1.5.
DIFFERENTIATION
OF
DISTRIBUTIONS
.
55
1.6.
MULTIPLICATION
OF
A
DISTRIBUTION
BY
A
SMOOTH
FUNCTION.
LINEAR
DIFFERENTIAL
OPERATORS
IN
SPACES
OF
DISTRIBUTIONS
.
57
1.7.
CHANGE
OF VARIABLES
AND
HOMOGENEOUS
DISTRIBUTIONS
.
58
1.8.
THE
DIRECT
OR
TENSOR
PRODUCT
OF
DISTRIBUTIONS
.
61
1.9.
THE
CONVOLUTION
OF
DISTRIBUTIONS
.
62
1.10.
THE
FOURIER
TRANSFORM
OF
TEMPERED
DISTRIBUTIONS
.
65
1.11.
THE
SCHWARTZ
KERNEL
OF
A
LINEAR
OPERATOR
.
68
1.12.
FUNDAMENTAL
SOLUTIONS
FOR
OPERATORS
WITH
CONSTANT
COEFFICIENTS
.
69
1.13.
A
FUNDAMENTAL
SOLUTION
FOR
THE
CAUCHY
PROBLEM
.
71
1.14.
FUNDAMENTAL
SOLUTIONS
AND
SOLUTIONS
OF
INHOMOGENEOUS
EQUATIONS
.
73
1.15.
DUHAMEL
'
S
PRINCIPLE
FOR
EQUATIONS
WITH
CONSTANT
COEFFICIENTS
.
75
1.16.
THE
FUNDAMENTAL
SOLUTION
AND
THE
BEHAVIOR
OF
SOLUTIONS
AT
INFINITY
.
77
1.17.
LOCAL
PROPERTIES
OF
SOLUTIONS
OF
HOMOGENEOUS
EQUATIONS
WITH
CONSTANT
COEFFICIENTS.
HYPOELLIPTICITY
AND
ELLIPTICITY
.
78
1.18.
LIOUVILLE
'
S
THEOREM
FOR
EQUATIONS
WITH
CONSTANT
COEFFICIENTS
.
80
1.19.
ISOLATED
SINGULARITIES
OF
SOLUTIONS
OF
HYPOELLIPTIC
EQUATIONS
.
81
§2.
ELLIPTIC
EQUATIONS
AND
BOUNDARY-VALUE
PROBLEMS
.
82
2.1.
THE
DEFINITION
OF
ELLIPTICITY.
THE
LAPLACE
AND
POISSON
EQUATIONS
.
82
CONTENTS
3
2.2.
A
FUNDAMENTAL
SOLUTION
FOR
THE
LAPLACIAN
OPERATOR.
GREEN
'
S
FORMULA
.
83
2.3.
MEAN-VALUE
THEOREMS
FOR
HARMONIC
FUNCTIONS
.
85
2.4.
THE
MAXIMUM
PRINCIPLE
FOR
HARMONIC
FUNCTIONS
AND
THE
NORMAL
DERIVATIVE
LEMMA
.
85
2.5.
UNIQUENESS
OF
THE
CLASSICAL
SOLUTIONS
OF
THE
DIRICHLET
AND
NEUMANN
PROBLEMS
FOR
LAPLACE
'
S
EQUATION
.
87
2.6.
INTERNAL
A
PRIORI
ESTIMATES
FOR
HARMONIC
FUNCTIONS.
HARNACK
'
S
THEOREM
.
87
2.7.
THE
GREEN
'
S
FUNCTION
OF
THE
DIRICHLET
PROBLEM
FOR
LAPLACE
'
S
EQUATION
.
88
2.8.
THE
GREEN
'
S
FUNCTION
AND
THE
SOLUTION
OF
THE
DIRICHLET
PROBLEM
FOR
A
BALL
AND
A
HALF-SPACE.
THE
REFLECTION
PRINCIPLE
.
90
2.9.
HARNACK
'
S
INEQUALITY
AND
LIOUVILLE
'
S
THEOREM
.
91
2.10.
THE
REMOVABLE
SINGULARITIES
THEOREM
.
92
2.11.
THE
KELVIN
TRANSFORM
AND
THE
STATEMENT
OF
EXTERIOR
BOUNDARY-VALUE
PROBLEMS
FOR
LAPLACE
'
S
EQUATION
.
92
2.12.
POTENTIALS
.
94
2.13.
APPLICATION
OF
POTENTIALS
TO
THE
SOLUTION
OF
BOUNDARY-VALUE
PROBLEMS
.
97
2.14.
BOUNDARY-VALUE
PROBLEMS
FOR
POISSON
'
S
EQUATION
IN
HOLDER
SPACES.
SCHAUDER
ESTIMATES
.
99
2.15.
CAPACITY
.
100
2.16.
THE
DIRICHLET
PROBLEM
IN
THE
CASE
OF
ARBITRARY
REGIONS
(THE
METHOD
OF
BALAYAGE).
REGULARITY
OF
A
BOUNDARY
POINT.
THE
WIENER
REGULARITY
CRITERION
.
102
2.17.
GENERAL
SECOND-ORDER
ELLIPTIC
EQUATIONS.
EIGENVALUES
AND
EIGENFUNCTIONS
OF
ELLIPTIC
OPERATORS
.
104
2.18.
HIGHER-ORDER
ELLIPTIC
EQUATIONS
AND
GENERAL
ELLIPTIC
BOUNDARY-VALUE
PROBLEMS.
THE
SHAPIRO-LOPATINSKIJ
CONDITION
.
105
2.19.
THE
INDEX
OF
AN
ELLIPTIC
BOUNDARY-VALUE
PROBLEM
.
110
2.20.
ELLIPTICITY
WITH
A
PARAMETER
AND
UNIQUE
SOLVABILITY
OF
ELLIPTIC
BOUNDARY-VALUE
PROBLEMS
.
ILL
§3.
SOBOLEV
SPACES
AND
GENERALIZED
SOLUTIONS
OF
BOUNDARY-VALUE
PROBLEMS
.
113
3.1.
THE
FUNDAMENTAL
SPACES
.
113
3.2.
IMBEDDING
AND
TRACE
THEOREMS
.
119
3.3.
GENERALIZED
SOLUTIONS
OF
ELLIPTIC
BOUNDARY-VALUE
PROBLEMS
AND
EIGENVALUE
PROBLEMS
.
122
3.4.
GENERALIZED
SOLUTIONS
OF
PARABOLIC
BOUNDARY-VALUE
PROBLEMS
.
132
3.5.
GENERALIZED
SOLUTIONS
OF
HYPERBOLIC
BOUNDARY-VALUE
PROBLEMS
.
134
4
CONTENTS
§4.
HYPERBOLIC
EQUATIONS
.
136
4.1.
DEFINITIONS
AND
EXAMPLES
.
136
4.2.
HYPERBOLICITY
AND
WELL-POSEDNESS
OF
THE
CAUCHY
PROBLEM
.
.
137
4.3.
ENERGY
ESTIMATES
.
138
4.4.
THE
SPEED
OF
PROPAGATION
OF
DISTURBANCES
.
141
4.5.
SOLUTION
OF
THE
CAUCHY
PROBLEM
FOR
THE
WAVE
EQUATION
.
.
.
141
4.6.
HUYGHENS
'
PRINCIPLE
.
144
4.7.
THE
PLANE
WAVE
METHOD
.
145
4.8.
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
IN
THE
PLANE
.
148
4.9.
LACUNAE
.
149
4.10.
THE
CAUCHY
PROBLEM
FOR
A
STRICTLY
HYPERBOLIC
SYSTEM
WITH
RAPIDLY
OSCILLATING
INITIAL
DATA
.
150
4.11.
DISCONTINUOUS
SOLUTIONS
OF
HYPERBOLIC
EQUATIONS
.
153
4.12.
SYMMETRIC
HYPERBOLIC
OPERATORS
.
157
4.13.
THE
MIXED
BOUNDARY-VALUE
PROBLEM
.
159
4.14.
THE
METHOD
OF
SEPARATION
OF
VARIABLES
.
162
§5.
PARABOLIC
EQUATIONS
.
163
5.1.
DEFINITIONS
AND
EXAMPLES
.
163
5.2.
THE
MAXIMUM
PRINCIPLE
AND
ITS
CONSEQUENCES
.
164
5.3.
INTEGRAL
ESTIMATES
.
166
5.4.
ESTIMATES
IN
HOLDER
SPACES
.
167
5.5.
THE
REGULARITY
OF
SOLUTIONS
OF
A
SECOND-ORDER
PARABOLIC
EQUATION
.
168
5.6.
POISSON
'
S
FORMULA
.
169
5.7.
A
FUNDAMENTAL
SOLUTION
OF
THE
CAUCHY
PROBLEM
FOR
A
SECOND-ORDER
EQUATION
WITH
VARIABLE
COEFFICIENTS
.
170
5.8.
SHILOV-PARABOLIC
SYSTEMS
.
172
5.9.
SYSTEMS
WITH
VARIABLE
COEFFICIENTS
.
173
5.10.
THE
MIXED
BOUNDARY-VALUE
PROBLEM
.
174
5.11.
STABILIZATION
OF
THE
SOLUTIONS
OF
THE
MIXED
BOUNDARY-VALUE
PROBLEM
AND
THE
CAUCHY
PROBLEM
.
176
§6.
GENERAL
EVOLUTION
EQUATIONS
.
177
6.1.
THE
CAUCHY
PROBLEM.
THE
HADAMARD
AND
PETROVSKIJ
CONDITIONS
.
177
6.2.
APPLICATION
OF
THE
LAPLACE
TRANSFORM
.
179
6.3.
APPLICATION
OF
THE
THEORY
OF
SEMIGROUPS
.
181
6.4.
SOME
EXAMPLES
.
183
§7.
EXTERIOR
BOUNDARY-VALUE
PROBLEMS
AND
SCATTERING
THEORY
.
.
.
184
7.1.
RADIATION
CONDITIONS
.
184
7.2.
THE
PRINCIPLE
OF
LIMITING
ABSORPTION
AND
LIMITING
AMPLITUDE
.
189
7.3.
RADIATION
CONDITIONS
AND
THE
PRINCIPLE
OF LIMITING
ABSORPTION
FOR
HIGHER-ORDER
EQUATIONS
AND
SYSTEMS
.
190
7.4.
DECAY
OF
THE
LOCAL
ENERGY
.
191
7.5.
SCATTERING
OF
PLANE
WAVES
.
192
CONTENTS
5
7.6.
SPECTRAL
ANALYSIS
.
193
7.7.
THE
SCATTERING
OPERATOR
AND
THE
SCATTERING
MATRIX
.
195
§8.
SPECTRAL
THEORY
OF
ONE-DIMENSIONAL
DIFFERENTIAL
OPERATORS
.
.
.
199
8.1.
OUTLINE
OF
THE
METHOD
OF
SEPARATION
OF
VARIABLES
.
199
8.2.
REGULAR
SELF-ADJOINT
PROBLEMS
.
201
8.3.
PERIODIC
AND
ANTIPERIODIC
BOUNDARY
CONDITIONS
.
206
8.4.
ASYMPTOTICS
OF
THE
EIGENVALUES
AND
EIGENFUNCTIONS
IN
THE
REGULAR
CASE
.
207
8.5.
THE
SCHRODINGER
OPERATOR
ON
A
HALF-LINE
.
210
8.6.
ESSENTIAL
SELF-ADJOINTNESS
AND
SELF-ADJOINT
EXTENSIONS.
THE
WEYL
CIRCLE
AND
THE
WEYL
POINT
.
211
8.7.
THE
CASE
OF
AN
INCREASING
POTENTIAL
.
214
8.8.
THE
CASE
OF
A
RAPIDLY
DECAYING
POTENTIAL
.
215
8.9.
THE
SCHRODINGER
OPERATOR
ON
THE
ENTIRE
LINE
.
216
8.10.
THE
HILL
OPERATOR
.
218
§9.
SPECIAL
FUNCTIONS
.
220
9.1.
SPHERICAL
FUNCTIONS
.
220
9.2.
THE
LEGENDRE
POLYNOMIALS
.
223
9.3.
CYLINDRICAL
FUNCTIONS
.
226
9.4.
PROPERTIES
OF
THE
CYLINDRICAL
FUNCTIONS
.
228
9.5.
AIRY
'
S
EQUATION
.
236
9.6.
SOME
OTHER
CLASSES
OF
FUNCTIONS
.
238
REFERENCES
.
242
AUTHOR
INDEX
.
248
SUBJECT
INDEX
.
251 |
| any_adam_object | 1 |
| author | Egorov, Jurij V. 1938- Šubin, Michail A. 1944- |
| author_GND | (DE-588)121177181 (DE-588)121177211 |
| author_facet | Egorov, Jurij V. 1938- Šubin, Michail A. 1944- |
| author_role | aut aut |
| author_sort | Egorov, Jurij V. 1938- |
| author_variant | j v e jv jve m a š ma maš |
| building | Verbundindex |
| bvnumber | BV011791582 |
| classification_rvk | SK 540 |
| classification_tum | MAT 350f |
| ctrlnum | (OCoLC)478751152 (DE-599)BVBBV011791582 |
| discipline | Mathematik |
| edition | 1. ed., 2. printing |
| format | Book |
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| id | DE-604.BV011791582 |
| illustrated | Not Illustrated |
| indexdate | 2024-08-14T01:12:07Z |
| institution | BVB |
| isbn | 3540638253 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007960363 |
| oclc_num | 478751152 |
| open_access_boolean | |
| owner | DE-703 DE-824 DE-384 DE-739 DE-91G DE-BY-TUM DE-83 DE-188 |
| owner_facet | DE-703 DE-824 DE-384 DE-739 DE-91G DE-BY-TUM DE-83 DE-188 |
| physical | 259 S. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| spelling | Egorov, Jurij V. 1938- Verfasser (DE-588)121177181 aut Foundations of the classical theory of partial differential equations Yu. V. Egorov ; M. A. Shubin 1. ed., 2. printing Berlin [u.a.] Springer 1998 259 S. txt rdacontent n rdamedia nc rdacarrier Erstdruck u.d.T.: Partial differential equations ; 1 Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Šubin, Michail A. 1944- Verfasser (DE-588)121177211 aut 1. Aufl. u.d.T. Encyclopaedia of mathematical sciences ; 30 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007960363&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Egorov, Jurij V. 1938- Šubin, Michail A. 1944- Foundations of the classical theory of partial differential equations Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 |
| title | Foundations of the classical theory of partial differential equations |
| title_auth | Foundations of the classical theory of partial differential equations |
| title_exact_search | Foundations of the classical theory of partial differential equations |
| title_full | Foundations of the classical theory of partial differential equations Yu. V. Egorov ; M. A. Shubin |
| title_fullStr | Foundations of the classical theory of partial differential equations Yu. V. Egorov ; M. A. Shubin |
| title_full_unstemmed | Foundations of the classical theory of partial differential equations Yu. V. Egorov ; M. A. Shubin |
| title_old | Encyclopaedia of mathematical sciences ; 30 |
| title_short | Foundations of the classical theory of partial differential equations |
| title_sort | foundations of the classical theory of partial differential equations |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007960363&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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