Weakly differentiable functions: Sobolev spaces and functions of bounded variation
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
[1989]
|
| Series: | Graduate texts in mathematics
120 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | xvi, 308 Seiten |
| ISBN: | 9781461269854 0387970177 3540970177 |
Staff View
MARC
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| 100 | 1 | |a Ziemer, William P. |d 1934-2017 |e Verfasser |0 (DE-588)172473683 |4 aut | |
| 245 | 1 | 0 | |a Weakly differentiable functions |b Sobolev spaces and functions of bounded variation |c William P. Ziemer |
| 264 | 1 | |a New York |b Springer |c [1989] | |
| 300 | |a xvi, 308 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 120 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 0 | 7 | |a Funktion von beschränkter Variation |0 (DE-588)4155666-5 |2 gnd |9 rswk-swf |
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| 689 | 1 | 0 | |a Sobolev-Raum |0 (DE-588)4055345-0 |D s |
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Record in the Search Index
| _version_ | 1804120338555994112 |
|---|---|
| adam_text | Contents
Preface
vii
1
Preliminaries
і
1.1
Notation
1
Inner product of vectors
Support of a function
Boundary of a set
Distance from a point to a set
Characteristic function of a set
Multi-indices
Partial derivative operators
Function spaces
—
continuous, Holder continuous,
Holder continuous derivatives
1.2
Measures onif
3
Lebesgue measurable sets
Lebesgue measurability of
Borei
sets
Suslin sets
1.3
Covering Theorems
7
Hausdorff maximal principle
General covering theorem
Vitali
covering theorem
Covering lemma, with
η
-balls whose radii vary in
Lipschitzian way
Besicovitch covering lemma
Besicovitch differentiation theorem
1.4
Hausdorff Measure
15
Equivalence of Hausdorff and Lebesgue measures
Hausdorff dimension
1.5
LP-Spaees
18
Integration of a function via its distribution
function
Young s inequality
Holder s and Jensen s inequality
1.6
Regularization
21
i^-spaces and
régularisation
xii Contents
1.7
Distributions
23
Functions and measures, as distributions
Positive distributions
Distributions determined by their local behavior
Convolution of distributions
Differentiation of distributions
1.8
Lorentz
Spaces
26
Non-increasing rearrangement of a function
Elementary properties of rearranged functions
Lorentz
spaces
O Neiľs
inequality, for rearranged functions
Equivalence of
LP-norm
and (p,p)-norm
Hardy s inequality
Inclusion relations of
Lorentz
spaces
Exercises
37
Historical Notes
39
2
Sobolev Spaces and Their Basic Properties
42
2.1
Weak Derivatives
42
Sobolev spaces
Absolute continuity on lines
ЈУ-погт
of difference quotients
Truncation of Sobolev functions
Composition of Sobolev functions
2.2
Change of Variables for Sobolev Functions
49
Rademacher s theorem
Bi-Lipschitzian change of variables
2.3
Approximation of Sobolev Functions by Smooth
Functions
53
Partition of unity
Smooth functions are dense in Wk p
2.4
Sobolev Inequalities
55
Sobolev s inequality
2.5
The Reffich-Kondrachov Compactness Theorem
61
Extension domains
2.6
Bessel Potentials and Capacity
64
Riesz and Bessel kernels
Bessel potentials
Bessel capacity
Basic properties of Bessel capacity
Capacitabiiity of Suslin sets
Minirnax theorem and alternate formulation of
Bessel capacity
Contents
Xlii
Metric
properties of Bessel capacity
2.7
The Best Constant in the Sobolev Inequality
76
Co-area formula
Sobolev s inequality and isoperimetric inequality
2.8
Alternate Proofs of the Fundamental Inequalities
83
Hardy-Littlewood-Wiener maximal theorem
Sobolev s inequality for Riesz potentials
2.9
Limiting Cases of the Sobolev Inequality
88
The case kp
=
η
by infinite series
The best constant in the case kp
=
η
An
ü/°°-bound in
the limiting case
2.10
Lorentz
Spaces, A Slight Improvement
96
Young s inequality in the context of
Lorentz
spaces
Sobolev s inequality in
Lorentz
spaces
The limiting case
Exercises
103
Historical Notes
108
3
Pointwise Behavior of Sobolev Functions
112
3.1
Limits of Integral Averages of Sobolev Functions
112
Limiting values of integral averages except for
capacity null set
3.2
Densities of Measures
116
3.3
Lebesgue Points for Sobolev Functions
118
Existence of Lebesgue points except for capacity
null set
Approximate continuity
Fine continuity everywhere except for capacity null set
3.4
¿P-Derivatives for Sobolev Functions
126
Existence of Taylor expansions
Iß
3.5
Properties of I^-Derivatives
130
The spaces Tk, tk,
Γ*·?, ί*·ρ
The implication of a function being in Tk v at all
points of a closed set
3.6
An IP-Version of the Whitney Extension Theorem
136
Existence of a C°° function comparable to the
distance function to a closed set
The Whitney extension theorem for functions in
Tk p and tk*
3.7
An Observation on Differentiation
142
3.8
Rademacher s Theorem in the IP-Context
145
A function in Tk p everywhere implies it is in
tk p almost everywhere
xjv Contents
3.9
The Implications of Pointwise Differentiability
146
Comparison of i^-derivatives and distributional
derivatives
If
u
Є
tk p(x) for every x, and if the
¿P-derivatives are in IP, then
и
Є
Wk v
3.10
A Lusin-Type Approximation for Sobolev Functions
153
Integral averages of Sobolev functions are uniformly
close to their limits on the complement of sets
of small capacity
Existence of smooth functions that agree with Sobolev
functions on the complement of sets of
small capacity
3.11
The Main Approximation
159
Existence of smooth functions that agree with
Sobolev functions on the complement of sets of
small capacity and are close in norm
Exercises
168
Historical Notes
175
4
Poincaré
Inequalities
—
A Unified Approach
177
4.1
Inequalities in a General Setting
178
An abstract version of the
Poincaré
inequality
4.2
Applications to Sobolev Spaces
182
An interpolation inequality
4.3
The Dual of Wm>P(ii)
185
The representation of {W™ p(u))*
4.4
Some Measures in
(W™*
(Ω))*
188
Poincaré
inequalities derived from the abstract
version by identifying Lebesgue and Hausdorff
measure with elements in
(И^т>р(О))*
The trace of Sobolev functions on the boundary of
Lipschitz domains
Poincaré
inequalities involving the trace of
a Sobolev function
4.5
Poincaré
Inequalities
193
Inequalities involving the capacity of the set on
which a function vanishes
4.6
Another Version of
Poincaré s
Inequality
196
An inequality involving dependence on the set on
which the function vanishes, not merely on its
capacity
4.7
More Measures in (Wm^(U))*
198
Sobolev s inequality for Riesz potentials involving
Contents xv
measures other than Lebesgue measure
Characterization of measures in (Wm>p{Rn))*
4.8
Other Inequalities Involving Measures in
(W*·*1)*
207
Inequalities involving the restriction of Hausdorff
measure to lower dimensional manifolds
4.9
The Case
ρ
= 1 209
Inequalities involving the L1-norm of the gradient
Exercises
214
Historical Notes
217
5
Functions of Bounded Variation
220
5.1
Definitions
220
Definition of BV functions
The total variation measure Du
5.2
Elementary Properties of BV Functions
222
Lower semicontmuity of the total variation measure
A condition ensuring continuity of the total
variation measure
5.3
Regularization of BV Functions
224
Regularization does not increase the BV norm
Approximation of BV functions by smooth functions
Compactness in Ll of the unit ball in BV
5.4
Sets of Finite Perimeter
228
Definition of sets of finite perimeter
The perimeter of domains with smooth boundaries
Isoperimetric and relative isoperimetric inequality for
sets of finite perimeter
5.5
The Generalized Exterior Normal
233
A preliminary version of the Gauss-Green theorem
Density results at points of the reduced boundary
5.6
Tangential Properties of the Reduced Boundary and the
Measure-Theoretic Normal
237
Blow-up at a point of the reduced boundary
The measure-theoretic normal
The reduced boundary is contained in the
measure-theoretic boundary
A lower bound for the density of
Ц-ОХбЦ
Hausdorff measure restricted to the reduced boundary
is bounded above by
jjűXsH
5.7
Rectifiability of the Reduced Boundary
243
Countably (n
-
lj-rectifiable sets
Countable
{» -
IJ-rectifiability of the
measure-theoretic boundary
xvi
Contents
5.8
The Gauss-Green Theorem
246
The equivalence of the restriction of Hausdorff
measure to the measure-theoretic boundary
and DXe
The Gauss-Green theorem for sets of finite perimeter
5.9
Pointwise Behavior of BV Functions
249
Upper and lower approximate limits
The Boxing inequality
The set of approximate jump discontinuities
5.10
The Trace of a BV Function
255
The bounded extension of BV functions
Trace of a BV function defined in terms of the
upper and lower approximate limits of the
extended function
The integrability of the trace over the
measure-theoretic boundary
5.11
Sobolev-Type Inequalities for BV Functions
260
Inequalities involving elements in
(ВК(П))*
5.12
Inequalities Involving Capacity
262
Characterization of measure in
(BV(íi))*
Poincaré
inequality for BV functions
5.13
Generalizations to the Case
ρ
> 1 270
5.14
Trace Defined in Terms of Integral Averages
272
Exercises
277
Historical Notes
280
Bibliography
283
List of Symbols
297
Index
303
|
| any_adam_object | 1 |
| author | Ziemer, William P. 1934-2017 |
| author_GND | (DE-588)172473683 |
| author_facet | Ziemer, William P. 1934-2017 |
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| building | Verbundindex |
| bvnumber | BV006114311 |
| classification_rvk | SK 600 |
| classification_tum | MAT 260f MAT 465f |
| ctrlnum | (OCoLC)231428019 (DE-599)BVBBV006114311 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV006114311 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:40:30Z |
| institution | BVB |
| isbn | 9781461269854 0387970177 3540970177 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003862671 |
| oclc_num | 231428019 |
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| physical | xvi, 308 Seiten |
| publishDate | 1989 |
| publishDateSearch | 1989 |
| publishDateSort | 1989 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Ziemer, William P. 1934-2017 Verfasser (DE-588)172473683 aut Weakly differentiable functions Sobolev spaces and functions of bounded variation William P. Ziemer New York Springer [1989] xvi, 308 Seiten txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 120 Hier auch später erschienene, unveränderte Nachdrucke Funktion von beschränkter Variation (DE-588)4155666-5 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 gnd rswk-swf Funktion von beschränkter Variation (DE-588)4155666-5 s DE-604 Sobolev-Raum (DE-588)4055345-0 s Erscheint auch als Online-Ausgabe, eBook 978-1-4612-1015-3 Graduate texts in mathematics 120 (DE-604)BV000000067 120 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003862671&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ziemer, William P. 1934-2017 Weakly differentiable functions Sobolev spaces and functions of bounded variation Graduate texts in mathematics Funktion von beschränkter Variation (DE-588)4155666-5 gnd Sobolev-Raum (DE-588)4055345-0 gnd |
| subject_GND | (DE-588)4155666-5 (DE-588)4055345-0 |
| title | Weakly differentiable functions Sobolev spaces and functions of bounded variation |
| title_auth | Weakly differentiable functions Sobolev spaces and functions of bounded variation |
| title_exact_search | Weakly differentiable functions Sobolev spaces and functions of bounded variation |
| title_full | Weakly differentiable functions Sobolev spaces and functions of bounded variation William P. Ziemer |
| title_fullStr | Weakly differentiable functions Sobolev spaces and functions of bounded variation William P. Ziemer |
| title_full_unstemmed | Weakly differentiable functions Sobolev spaces and functions of bounded variation William P. Ziemer |
| title_short | Weakly differentiable functions |
| title_sort | weakly differentiable functions sobolev spaces and functions of bounded variation |
| title_sub | Sobolev spaces and functions of bounded variation |
| topic | Funktion von beschränkter Variation (DE-588)4155666-5 gnd Sobolev-Raum (DE-588)4055345-0 gnd |
| topic_facet | Funktion von beschränkter Variation Sobolev-Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003862671&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT ziemerwilliamp weaklydifferentiablefunctionssobolevspacesandfunctionsofboundedvariation |