Lectures on elliptic and parabolic equations in Sobolev spaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2008
|
| Series: | Graduate studies in mathematics
96 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XVIII, 357 S. |
| ISBN: | 9780821846841 |
Staff View
MARC
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| 100 | 1 | |a Krylov, Nikolaj V. |d 1941- |e Verfasser |0 (DE-588)121699684 |4 aut | |
| 245 | 1 | 0 | |a Lectures on elliptic and parabolic equations in Sobolev spaces |c N.V. Krylov |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2008 | |
| 300 | |a XVIII, 357 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 96 | |
| 650 | 4 | |a Sobolev, Espaces de | |
| 650 | 4 | |a Équations différentielles elliptiques | |
| 650 | 4 | |a Équations différentielles paraboliques | |
| 650 | 4 | |a Differential equations, Elliptic | |
| 650 | 4 | |a Differential equations, Parabolic | |
| 650 | 4 | |a Sobolev spaces | |
| 650 | 0 | 7 | |a Sobolev-Raum |0 (DE-588)4055345-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Parabolische Differentialgleichung |0 (DE-588)4173245-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |2 gnd |9 rswk-swf |
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| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-2121-2 |
| 830 | 0 | |a Graduate studies in mathematics |v 96 |w (DE-604)BV009739289 |9 96 | |
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| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754925&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
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Record in the Search Index
| _version_ | 1804138041786236928 |
|---|---|
| adam_text | Contents
Preface
xi
Chapter
1.
Second-order elliptic equations in W|
(lĄ
1
§1.
The simplest equation Xu
—
Au
= ƒ 2
§2.
Integrating the determinants of Hessians (optional)
7
§3.
Sobolev spaces W£(Sl)
8
§4.
Second-order elliptic differential operators
14
§5.
Multiplicative inequalities
17
§6.
Solvability of elliptic equations with continuous coefficients
20
§7.
Higher regularity of solutions
25
§8.
Sobolev mollifiers
32
§9.
Singular-integral representation of uxx
38
§10.
Hints to exercises
42
Chapter
2.
Second-order parabolic equations in W^M * 1 1)
45
§1.
The simplest equation
щ
+
al^(t)DijU
—
Xu =
ƒ 45
§2.
Sobolev spaces W£k+2r(U)
51
§3.
Parabolic equations with continuous coefficients
56
§4.
Local or interior estimates
60
§5.
The Cauchy problem
67
vi
Contents
§6.
Hints to exercises
70
Chapter
3.
Some tools from real analysis
73
§1.
Partitions and stopping times
73
§2.
Maximal and sharp functions
78
2:1.
The Fefferman-Stein theorem
78
2:2.
Exercises (optional)
83
§3.
Comparing maximal and sharp functions in the Euclidean
space
87
§4.
Hints to exercises
91
Chapter
4.
Basic Cp estimates for parabolic and elliptic equations
93
§1.
An approach to elliptic equations
94
§2.
Preliminary estimates of L-caloric functions
98
§3.
Solvability of model equations
104
§4.
Divergence form of the right-hand side for the Laplacian
113
§5.
Hints to exercises
115
Chapter
5.
Parabolic and elliptic equations in Wp and
W*
117
§1.
Better regularity for equations with coefficients
independent of
χ
117
§2.
Equations with continuous coefficients. The Cauchy problem
119
Chapter
6.
Equations with VMO coefficients
125
§1.
Estimating Cq oscillations of uxx
125
§2.
Estimating sharp functions of uxx
130
§3.
A priori estimates for parabolic and elliptic equations with
VMO coefficients
134
§4.
Solvability of parabolic and elliptic equations with
VMO coefficients. The Cauchy problem
139
§5.
Hints to exercises
143
Chapter
7.
Parabolic equations with VMO coefficients in spaces with
mixed norms
145
§1.
Estimating sharp functions of
||гіхя.(г,
-) r
146
Contents
vii
§2.
Existence
and uniqueness results
149
§3.
Hints to exercises
155
Chapter
8.
Second-order elliptic equations in
W£(d)
157
§1.
Spaces of functions vanishing on the boundary
158
§2.
Equations in half spaces
160
§3.
Domains of class Ck. Equations near the boundary
165
§4.
Partitions of unity and the regularizer
171
§5.
Solvability of equations in domains for large
λ
174
§6.
Hints to exercises
179
Chapter
9.
Second-order elliptic equations in W£(u)
181
§1.
Finite differences. Better regularity of solutions in R^
for model equations
181
§2.
Equations in domains
187
§3.
The oblique derivative problem in R^
190
§4.
Local regularity of solutions
196
§5.
Hints to exercises
199
Chapter
10.
Sobolev embedding theorems for W^(O)
201
§1.
Embedding for
Campanaio
and Slobodetskii spaces
203
1:1.
Embeddings in Ca
203
1:2.
Exercises (optional)
207
§2.
Embedding
W$(ß)
С С1 ^«).
Morrey s theorem
209
§3.
The Gagliardo-Nirenberg theorem
215
§4.
General embedding theorems
216
§5.
Compactness of embeddings. Kondrashov s theorem
223
§6.
An application of Riesz s theory of compact operators
227
§7.
Hints to exercises
229
Chapter
11.
Second-order elliptic equations
Lu
—
Хи =
ƒ
with
λ
small
231
§1.
Maximum principle for smooth functions
232
1:1.
Maximum principle
232
viii Contents
1:2.
Exercises
(optional) 234
§2. Resolvent
operator
for
λ
large
236
§3.
Solvability of equations in smooth domains
241
§4.
The way we proceed further
245
§5.
Decay at infinity of solutions of
Lu
= ƒ
in Rd
246
§6.
Equations in Md with
λ
small
249
§7.
Traces of W£(u) functions on
дп
255
§8.
The maximum principle in Wp
.
Green s functions
262
§9.
Hints to exercises
263
Chapter
12.
Fourier transform and elliptic operators
267
§1.
The space
S
268
§2.
The notion of elliptic differential operator
269
§3.
Comments on the oblique derivative and other
boundary-value problems. Instances of pseudo-differential
operators
272
§4.
Pseudo-differential operators
274
§5.
Green s functions
280
§6.
Existence of Green s functions
285
§7.
Estimating
G
and its derivatives
288
7:1.
Differentiability of
G
and estimates of its derivatives
288
7:2.
Exercises (optional)
292
§8.
Boundedness of the zeroth-order pseudo-differential
operators in £p
293
§9.
Operators related to the Laplacian
297
9:1.
The operators
(1 —
Δ)7/2,
Cauchy s operator, the Riesz
and Hubert transforms, the Cauchy-Riemann operator
297
9:2.
Exercises (optional)
302
§10.
An embedding lemma
304
§11.
Hints to exercises
308
Chapter
13.
Elliptic operators and the spaces Hp
311
§1.
The space
Η
311
Contents ix
§2.
Some properties of the space
Ή
315
§3.
The spaces
Щ
317
3:1.
Definition, solvability of elliptic equations in Hp, the
equality
Щ
=
Wp7
317
3:2.
Exercises (optional)
323
§4.
Higher-order elliptic differential equations with
continuous coefficients in Hp
328
§5.
Second-order parabolic equations. Semigroups (optional)
333
§6.
Second-order divergence type elliptic equations with
continuous coefficients
335
§7.
Nonzero Dirichlet condition and traces
339
§8.
Sobolev embedding theorems for
Щ
spaces
342
§9.
Sobolev mollifiers
346
§10.
Hints to exercises
350
Bibliography
353
Index
355
This book concentrates on the basic facts and ideas of the modern theory of linear
elliptic and parabolic equations in Sobolev spaces.
The main areas covered in this book are the first boundary-value problem for
elliptic equations and the Cauchy problem for parabolic equations. In addition, other
boundary-value problems such as the Neumann or oblique derivative problems are
briefly covered. As is natural for a textbook, the main emphasis is on organizing well-
known ideas in a self-contained exposition. Among the topics included that are not
usually covered in a textbook are a relatively recent development concerning equa¬
tions with VMO coefficients and the study of parabolic equations with coefficients
measurable only with respect to the time variable. There are numerous exercises
which help the reader better understand the material.
After going through the book, the reader will have a good understanding of results
available in the modern theory of partial differential equations and the technique used
to obtain them. Prerequisites are basics of measure theory, the theory of Lp spaces,
and the Fourier transform.
|
| adam_txt |
Contents
Preface
xi
Chapter
1.
Second-order elliptic equations in W|
(lĄ
1
§1.
The simplest equation Xu
—
Au
= ƒ 2
§2.
Integrating the determinants of Hessians (optional)
7
§3.
Sobolev spaces W£(Sl)
8
§4.
Second-order elliptic differential operators
14
§5.
Multiplicative inequalities
17
§6.
Solvability of elliptic equations with continuous coefficients
20
§7.
Higher regularity of solutions
25
§8.
Sobolev mollifiers
32
§9.
Singular-integral representation of uxx
38
§10.
Hints to exercises
42
Chapter
2.
Second-order parabolic equations in W^M'*"1"1)
45
§1.
The simplest equation
щ
+
al^(t)DijU
—
Xu =
ƒ 45
§2.
Sobolev spaces W£k+2r(U)
51
§3.
Parabolic equations with continuous coefficients
56
§4.
Local or interior estimates
60
§5.
The Cauchy problem
67
vi
Contents
§6.
Hints to exercises
70
Chapter
3.
Some tools from real analysis
73
§1.
Partitions and stopping times
73
§2.
Maximal and sharp functions
78
2:1.
The Fefferman-Stein theorem
78
2:2.
Exercises (optional)
83
§3.
Comparing maximal and sharp functions in the Euclidean
space
87
§4.
Hints to exercises
91
Chapter
4.
Basic Cp estimates for parabolic and elliptic equations
93
§1.
An approach to elliptic equations
94
§2.
Preliminary estimates of L-caloric functions
98
§3.
Solvability of model equations
104
§4.
Divergence form of the right-hand side for the Laplacian
113
§5.
Hints to exercises
115
Chapter
5.
Parabolic and elliptic equations in Wp' and
W*
117
§1.
Better regularity for equations with coefficients
independent of
χ
117
§2.
Equations with continuous coefficients. The Cauchy problem
119
Chapter
6.
Equations with VMO coefficients
125
§1.
Estimating Cq oscillations of uxx
125
§2.
Estimating sharp functions of uxx
130
§3.
A priori estimates for parabolic and elliptic equations with
VMO coefficients
134
§4.
Solvability of parabolic and elliptic equations with
VMO coefficients. The Cauchy problem
139
§5.
Hints to exercises
143
Chapter
7.
Parabolic equations with VMO coefficients in spaces with
mixed norms
145
§1.
Estimating sharp functions of
||гіхя.(г,
-)\\r
146
Contents
vii
§2.
Existence
and uniqueness results
149
§3.
Hints to exercises
155
Chapter
8.
Second-order elliptic equations in
W£(d)
157
§1.
Spaces of functions vanishing on the boundary
158
§2.
Equations in half spaces
160
§3.
Domains of class Ck. Equations near the boundary
165
§4.
Partitions of unity and the regularizer
171
§5.
Solvability of equations in domains for large
λ
174
§6.
Hints to exercises
179
Chapter
9.
Second-order elliptic equations in W£(u)
181
§1.
Finite differences. Better regularity of solutions in R^
for model equations
181
§2.
Equations in domains
187
§3.
The oblique derivative problem in R^
190
§4.
Local regularity of solutions
196
§5.
Hints to exercises
199
Chapter
10.
Sobolev embedding theorems for W^(O)
201
§1.
Embedding for
Campanaio
and Slobodetskii spaces
203
1:1.
Embeddings in Ca
203
1:2.
Exercises (optional)
207
§2.
Embedding
W$(ß)
С С1"^«).
Morrey's theorem
209
§3.
The Gagliardo-Nirenberg theorem
215
§4.
General embedding theorems
216
§5.
Compactness of embeddings. Kondrashov's theorem
223
§6.
An application of Riesz's theory of compact operators
227
§7.
Hints to exercises
229
Chapter
11.
Second-order elliptic equations
Lu
—
Хи =
ƒ
with
λ
small
231
§1.
Maximum principle for smooth functions
232
1:1.
Maximum principle
232
viii Contents
1:2.
Exercises
(optional) 234
§2. Resolvent
operator
for
λ
large
236
§3.
Solvability of equations in smooth domains
241
§4.
The way we proceed further
245
§5.
Decay at infinity of solutions of
Lu
= ƒ
in Rd
246
§6.
Equations in Md with
λ
small
249
§7.
Traces of W£(u) functions on
дп
255
§8.
The maximum principle in Wp
.
Green's functions
262
§9.
Hints to exercises
263
Chapter
12.
Fourier transform and elliptic operators
267
§1.
The space
S
268
§2.
The notion of elliptic differential operator
269
§3.
Comments on the oblique derivative and other
boundary-value problems. Instances of pseudo-differential
operators
272
§4.
Pseudo-differential operators
274
§5.
Green's functions
280
§6.
Existence of Green's functions
285
§7.
Estimating
G
and its derivatives
288
7:1.
Differentiability of
G
and estimates of its derivatives
288
7:2.
Exercises (optional)
292
§8.
Boundedness of the zeroth-order pseudo-differential
operators in £p
293
§9.
Operators related to the Laplacian
297
9:1.
The operators
(1 —
Δ)7/2,
Cauchy's operator, the Riesz
and Hubert transforms, the Cauchy-Riemann operator
297
9:2.
Exercises (optional)
302
§10.
An embedding lemma
304
§11.
Hints to exercises
308
Chapter
13.
Elliptic operators and the spaces Hp
311
§1.
The space
Η
311
Contents ix
§2.
Some properties of the space
Ή
315
§3.
The spaces
Щ
317
3:1.
Definition, solvability of elliptic equations in Hp, the
equality
Щ
=
Wp7
317
3:2.
Exercises (optional)
323
§4.
Higher-order elliptic differential equations with
continuous coefficients in Hp
328
§5.
Second-order parabolic equations. Semigroups (optional)
333
§6.
Second-order divergence type elliptic equations with
continuous coefficients
335
§7.
Nonzero Dirichlet condition and traces
339
§8.
Sobolev embedding theorems for
Щ
spaces
342
§9.
Sobolev mollifiers
346
§10.
Hints to exercises
350
Bibliography
353
Index
355
This book concentrates on the basic facts and ideas of the modern theory of linear
elliptic and parabolic equations in Sobolev spaces.
The main areas covered in this book are the first boundary-value problem for
elliptic equations and the Cauchy problem for parabolic equations. In addition, other
boundary-value problems such as the Neumann or oblique derivative problems are
briefly covered. As is natural for a textbook, the main emphasis is on organizing well-
known ideas in a self-contained exposition. Among the topics included that are not
usually covered in a textbook are a relatively recent development concerning equa¬
tions with VMO coefficients and the study of parabolic equations with coefficients
measurable only with respect to the time variable. There are numerous exercises
which help the reader better understand the material.
After going through the book, the reader will have a good understanding of results
available in the modern theory of partial differential equations and the technique used
to obtain them. Prerequisites are basics of measure theory, the theory of Lp spaces,
and the Fourier transform. |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Krylov, Nikolaj V. 1941- |
| author_GND | (DE-588)121699684 |
| author_facet | Krylov, Nikolaj V. 1941- |
| author_role | aut |
| author_sort | Krylov, Nikolaj V. 1941- |
| author_variant | n v k nv nvk |
| building | Verbundindex |
| bvnumber | BV035086751 |
| callnumber-first | Q - Science |
| callnumber-label | QA377 |
| callnumber-raw | QA377 |
| callnumber-search | QA377 |
| callnumber-sort | QA 3377 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 600 SK 560 |
| ctrlnum | (OCoLC)222250824 (DE-599)BVBBV035086751 |
| dewey-full | 515/.3533 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.3533 |
| dewey-search | 515/.3533 |
| dewey-sort | 3515 43533 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035086751 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T22:09:11Z |
| indexdate | 2024-07-09T21:21:53Z |
| institution | BVB |
| isbn | 9780821846841 |
| language | English |
| lccn | 2008016051 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016754925 |
| oclc_num | 222250824 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-824 DE-703 DE-83 DE-188 DE-19 DE-BY-UBM DE-739 DE-20 DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-824 DE-703 DE-83 DE-188 DE-19 DE-BY-UBM DE-739 DE-20 DE-11 |
| physical | XVIII, 357 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Krylov, Nikolaj V. 1941- Verfasser (DE-588)121699684 aut Lectures on elliptic and parabolic equations in Sobolev spaces N.V. Krylov Providence, RI American Math. Soc. 2008 XVIII, 357 S. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 96 Sobolev, Espaces de Équations différentielles elliptiques Équations différentielles paraboliques Differential equations, Elliptic Differential equations, Parabolic Sobolev spaces Sobolev-Raum (DE-588)4055345-0 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 s Sobolev-Raum (DE-588)4055345-0 s DE-604 Elliptische Differentialgleichung (DE-588)4014485-9 s Erscheint auch als Online-Ausgabe 978-1-4704-2121-2 Graduate studies in mathematics 96 (DE-604)BV009739289 96 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754925&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754925&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Krylov, Nikolaj V. 1941- Lectures on elliptic and parabolic equations in Sobolev spaces Graduate studies in mathematics Sobolev, Espaces de Équations différentielles elliptiques Équations différentielles paraboliques Differential equations, Elliptic Differential equations, Parabolic Sobolev spaces Sobolev-Raum (DE-588)4055345-0 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| subject_GND | (DE-588)4055345-0 (DE-588)4173245-5 (DE-588)4014485-9 |
| title | Lectures on elliptic and parabolic equations in Sobolev spaces |
| title_auth | Lectures on elliptic and parabolic equations in Sobolev spaces |
| title_exact_search | Lectures on elliptic and parabolic equations in Sobolev spaces |
| title_exact_search_txtP | Lectures on elliptic and parabolic equations in Sobolev spaces |
| title_full | Lectures on elliptic and parabolic equations in Sobolev spaces N.V. Krylov |
| title_fullStr | Lectures on elliptic and parabolic equations in Sobolev spaces N.V. Krylov |
| title_full_unstemmed | Lectures on elliptic and parabolic equations in Sobolev spaces N.V. Krylov |
| title_short | Lectures on elliptic and parabolic equations in Sobolev spaces |
| title_sort | lectures on elliptic and parabolic equations in sobolev spaces |
| topic | Sobolev, Espaces de Équations différentielles elliptiques Équations différentielles paraboliques Differential equations, Elliptic Differential equations, Parabolic Sobolev spaces Sobolev-Raum (DE-588)4055345-0 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| topic_facet | Sobolev, Espaces de Équations différentielles elliptiques Équations différentielles paraboliques Differential equations, Elliptic Differential equations, Parabolic Sobolev spaces Sobolev-Raum Parabolische Differentialgleichung Elliptische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754925&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754925&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT krylovnikolajv lecturesonellipticandparabolicequationsinsobolevspaces |