An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Pisa
Ed. della Normale
2005
|
| Schriftenreihe: | Appunti
2 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | IX, 302 |
| ISBN: | 9788876421686 |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
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| 003 | DE-604 | ||
| 005 | 20201012 | ||
| 007 | t | ||
| 008 | 101208s2005 |||| 00||| eng d | ||
| 020 | |a 9788876421686 |9 978-88-7642-168-6 | ||
| 020 | |z 8876421688 |9 88-7642-168-8 | ||
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| 041 | 0 | |a eng | |
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| 100 | 1 | |a Giaquinta, Mariano |d 1947- |e Verfasser |0 (DE-588)111595738 |4 aut | |
| 245 | 1 | 0 | |a An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs |c Mariano Giaquinta ; Luca Martinazzi |
| 264 | 1 | |a Pisa |b Ed. della Normale |c 2005 | |
| 300 | |a IX, 302 | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Appunti |v 2 | |
| 650 | 0 | 7 | |a Regularität |0 (DE-588)4049074-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Harmonische Abbildung |0 (DE-588)4023452-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptisches System |0 (DE-588)4121184-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Minimaler Graph |0 (DE-588)4873867-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Regularität |0 (DE-588)4049074-9 |D s |
| 689 | 0 | 1 | |a Elliptisches System |0 (DE-588)4121184-4 |D s |
| 689 | 0 | 2 | |a Minimaler Graph |0 (DE-588)4873867-0 |D s |
| 689 | 0 | 3 | |a Harmonische Abbildung |0 (DE-588)4023452-6 |D s |
| 689 | 0 | |8 1\p |5 DE-604 | |
| 689 | 1 | 0 | |a Regularität |0 (DE-588)4049074-9 |D s |
| 689 | 1 | 1 | |a Minimaler Graph |0 (DE-588)4873867-0 |D s |
| 689 | 1 | |5 DE-604 | |
| 689 | 2 | 0 | |a Regularität |0 (DE-588)4049074-9 |D s |
| 689 | 2 | 1 | |a Harmonische Abbildung |0 (DE-588)4023452-6 |D s |
| 689 | 2 | |5 DE-604 | |
| 700 | 1 | |a Martinazzi, Luca |d 1981- |e Verfasser |0 (DE-588)1138459216 |4 aut | |
| 830 | 0 | |a Appunti |v 2 |w (DE-604)BV025597622 |9 2 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020773709&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-020773709 | ||
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
Datensatz im Suchindex
| _version_ | 1804143555832184832 |
|---|---|
| adam_text | Contents
Preface
XI
1 Harmonie
functions
1
Harmonie
functions
1
1.1.
Introduction
........................ 1
1.2.
The variational method
.................. 2
1.2.1.
Non-existence of minimizers
........... 3
1.2.2.
Non-finiteness of the Dirichlet integral
...... 4
1.3.
Some properties of harmonic functions
.......... 5
1.4.
Existence in general bounded domains
.......... 10
1.4.1.
Solvability of the Dirichlet problem on balls:
Poisson s formula
................. 10
1.4.2.
Perron s method
................. 11
1.4.3.
Poincaré s
method
................ 13
2
Direct methods
15
2.1.
Lower semicontinuity
................... 16
2.2.
Existence of minimizers
................. 17
2.2.1.
Minimizers in
Lip¿(n)
.............. 17
2.2.2.
A priori gradient estimates
............ 18
2.2.3.
Constructing barriers: the distance function
... 21
2.3.
Non-existence
....................... 22
2.3.1.
An example of Bernstein
............. 22
2.3.2.
Sharpness of the mean curvature condition
... 24
2.4.
Finiteness of the area of graphs with zero mean curvature
25
2.5.
The area functional in BV
................ 26
2.5.1.
B V minimixers
for the area functional
. ..... 27
VI
Mariano
Giaquinta
buca
Martina/ži
ì Hilbert
space methods 31
3.1.
The Dirichlet principle
.................. 31
3.2.
Sobolev spaces
...................... 33
3.2.1.
Strong and weak derivatives
........... 33
3.2.2.
Poincaré
inequalities
............... 34
3.2.3.
Rellich s theorem
................. 36
3.2.4.
The Sobolev embedding theorem
........ 38
3.3.
Elliptic equations: existence of weak solutions
...... 38
3.3.1.
Dirichlet boundary condition
........... 38
3.3.2.
Neumann boundary condition
.......... 40
3.4.
Elliptic systems: existence of weak solutions
...... 42
3.4.1.
The Legendre and Legendre-Hadamard
ellipticity conditions
............... 42
3.4.2.
Boundary value problems for very strongly
elliptic systems
.................. 43
3.4.3.
Strongly elliptic systems: Garding s inequality
. 44
4
L2 regularity: the Caccioppoli inequality
49
4.1.
The simplest case: harmonic functions
.......... 49
4.2.
Caccioppoli s inequality for elliptic systems
....... 51
4.3.
The difference quotient method
.............. 52
4.3.1.
Interior L2-estimates
............... 54
4.3.2.
Boundary regularity
............... 57
4.4.
The hole-filling technique
................. 58
5 Schauder
estimates
61
5.1.
The spaces of Morrey and
Campanaio
.......... 61
5.1.1.
A characterization of Holder continuous functions
63
5.2.
Constant coefficients: two basic estimates
........ 65
5.2.1.
A generalization of Liouville s theorem
..... 67
5.3.
A lemma
.......................... 68
5.4. Schauder
estimates for systems in divergence form
... 69
5.4.1.
Constant coefficients
............... 69
5.4.2.
Continuous coefficients
.............. 71
5.4.3.
Holder continuous coefficients
.......... 72
5.4.4.
Summary and generalizations
.......... 73
5.4.5.
Boundary regularity
............... 74
5.5. Schauder
estimates for systems in non-divergence form
. 77
5.5.1.
Solving the Dirichlet problem
.......... 79
VII
Ли
introduction to the regularity theory
6
Some real analysis
83
6.1.
Distribution function and Marcinkiewicz interpolation
. . 83
6.1.1.
The distribution function
............. 83
6.1.2.
Riesz-Thorin s theorem
.............. 85
6.1.3.
Marcinkiewicz s interpolation theorem
..... 87
6.2.
The maximal function and Calderon-Zygmund argument
89
6.2.1.
The maximal function
.............. 89
6.2.2.
Calderon-Zygmund decomposition argument
. . 94
6.3.
BMO
........................... 96
6.3.1.
John-Nirenberg space
............... 96
6.3.2.
John-Nirenberg lemma I
............. 97
6.3.3.
John-Nirenberg lemma II
............. 102
6.3.4.
Interpolation between Lp and BMO
....... 106
6.3.5.
Sharp function and interpolation Lp
-
BMO
. . 108
6.4.
Reverse Holder inequalities
................
Ill
6.4.1.
Gehring s lemma
.................
Ill
6.4.2.
Reverse Holder inequalities with
increasing support
................ 114
7
LP-theory
117
7.1.
Lp-estimates
....................... 117
7.1.1.
Constant coefficients
............... 117
7.1.2.
Variable coefficients:
divergence and non-divergence case
....... 118
7.2.
Singular integrals
..................... 120
7.2.1. Hölder-Korn-Lichtenstein-Giraud
theorem
... 125
7.2.2.
L2-theory
..................... 128
7.2.3.
Calderon-Zygmund theorem
........... 131
7.3.
Fractional integrals and Sobolev inequalities
....... 137
8
The regularity problem in the scalar case
143
8.1.
Existence by direct methods
............... 143
8.2.
Regularity of critical points of variational integrals
... 146
8.3. De
Giorgi s theorem: essentially the original proof
. . . 149
8.4.
Moser s technique and Harnack s inequality
....... 158
8.5.
Still another proof of
De
Giorgi s theorem
........ 163
8.6.
The weak Harnack inequality
............... 167
8.7.
Non-differentiabie variational integrals
.......... 170
VIII
Murían»
Cìiaqiiinlu
-
Luca
Martina™
9
Partial
regularity in the vector-valued case
175
9.1.
Counterexamples to everywhere regularity
........ 175
9.2.
Partial regularity
..................... 177
9.2.1.
Partial regularity of minimizers
......... 177
9.2.2.
Partial regularity of solutions
to
quasilinear
elliptic systems
.......... 180
9.2.3.
Partial regularity of solutions
to
quasilinear
elliptic systems with quadratic right-
hand side
..................... 184
9.2.4.
Partial regularity of minimizers of
non-differentiable
quadratic functionals
............... 190
9.2.5.
The Hausdorff dimension of the singular set
. . . 195
10
Harmonic maps
199
10.1.
Basic material
....................... 199
10.1.1.
The variational equations
............. 200
10.1.2.
The
monotonicity
formula
............ 202
10.2.
Giaquinta and Giusti s regularity results
......... 203
10.2.1.
The main regularity result
............ 203
10.2.2.
The dimension reduction argument
....... 204
10.3.
Schoen
and Uhlenbeck s regularity results
........ 211
10.3.1.
The main regularity result
............ 211
10.3.2.
The dimension reduction argument
....... 219
10.3.3.
The stratification of the singular set
....... 229
11
A survey of minimal graphs
235
11.1.
Geometry of the submanifolds of W+m
......... 235
11.1.1.
Riemannian structure and Levi-Civita connection
235
11.1.2.
Second fundamental form and mean curvature
. . 238
11.1.3.
First variation of the area
............. 240
11.1.4.
Area-decreasing maps
.............. 245
11.2.
Minimal graphs in codimension
1 ............ 246
11.2.1.
Convexity of the area; uniqueness and stability
. 246
11.2.2.
The problem of Plateau: existence of minimal
graphs with prescribed boundary
......... 248
11.2.3.
A priori estimates
................. 251
11.2.4.
Regularity of Lipschitz continuous
minimal graphs
.................. 255
11.2.5.
The a priori gradient estimate of Bombieri,
De Giorgi
and Miranda
.............. 255
IX An introduction to the regularity theory
11.2.6.
Regularity of
В
V minimizers of
the area functional
................ 259
11.3.
Regularity in arbitrary codimension
........... 262
11.3.1.
The theorem of Allard
.............. 263
11.3.2.
Blow-ups and blow-downs: minimal cones
. . . 264
11.3.3.
Bernstein-type theorems
............. 269
11.3.4.
Regularity of area-decreasing minimal graphs
. . 276
11.4.
Geometry of Varifolds
.................. 278
11.4.1.
Rectifiable subsets of Rn+m
........... 278
11.4.2.
Rectifiable varifolds
............... 281
11.4.3.
Abstract varifolds
................. 285
11.4.4.
Allard s compactness theorem
.......... 288
References
291
Index
297
|
| any_adam_object | 1 |
| author | Giaquinta, Mariano 1947- Martinazzi, Luca 1981- |
| author_GND | (DE-588)111595738 (DE-588)1138459216 |
| author_facet | Giaquinta, Mariano 1947- Martinazzi, Luca 1981- |
| author_role | aut aut |
| author_sort | Giaquinta, Mariano 1947- |
| author_variant | m g mg l m lm |
| building | Verbundindex |
| bvnumber | BV036857880 |
| classification_rvk | SK 660 |
| ctrlnum | (OCoLC)254969532 (DE-599)BVBBV036857880 |
| dewey-full | 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 |
| dewey-search | 510 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV036857880 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:49:32Z |
| institution | BVB |
| isbn | 9788876421686 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020773709 |
| oclc_num | 254969532 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR |
| owner_facet | DE-355 DE-BY-UBR |
| physical | IX, 302 |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Ed. della Normale |
| record_format | marc |
| series | Appunti |
| series2 | Appunti |
| spelling | Giaquinta, Mariano 1947- Verfasser (DE-588)111595738 aut An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs Mariano Giaquinta ; Luca Martinazzi Pisa Ed. della Normale 2005 IX, 302 txt rdacontent n rdamedia nc rdacarrier Appunti 2 Regularität (DE-588)4049074-9 gnd rswk-swf Harmonische Abbildung (DE-588)4023452-6 gnd rswk-swf Elliptisches System (DE-588)4121184-4 gnd rswk-swf Minimaler Graph (DE-588)4873867-0 gnd rswk-swf Regularität (DE-588)4049074-9 s Elliptisches System (DE-588)4121184-4 s Minimaler Graph (DE-588)4873867-0 s Harmonische Abbildung (DE-588)4023452-6 s 1\p DE-604 DE-604 Martinazzi, Luca 1981- Verfasser (DE-588)1138459216 aut Appunti 2 (DE-604)BV025597622 2 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020773709&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Giaquinta, Mariano 1947- Martinazzi, Luca 1981- An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs Appunti Regularität (DE-588)4049074-9 gnd Harmonische Abbildung (DE-588)4023452-6 gnd Elliptisches System (DE-588)4121184-4 gnd Minimaler Graph (DE-588)4873867-0 gnd |
| subject_GND | (DE-588)4049074-9 (DE-588)4023452-6 (DE-588)4121184-4 (DE-588)4873867-0 |
| title | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs |
| title_auth | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs |
| title_exact_search | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs |
| title_full | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs Mariano Giaquinta ; Luca Martinazzi |
| title_fullStr | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs Mariano Giaquinta ; Luca Martinazzi |
| title_full_unstemmed | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs Mariano Giaquinta ; Luca Martinazzi |
| title_short | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs |
| title_sort | an introduction to the regularity theory for elliptic systems harmonic maps and minimal graphs |
| topic | Regularität (DE-588)4049074-9 gnd Harmonische Abbildung (DE-588)4023452-6 gnd Elliptisches System (DE-588)4121184-4 gnd Minimaler Graph (DE-588)4873867-0 gnd |
| topic_facet | Regularität Harmonische Abbildung Elliptisches System Minimaler Graph |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020773709&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV025597622 |
| work_keys_str_mv | AT giaquintamariano anintroductiontotheregularitytheoryforellipticsystemsharmonicmapsandminimalgraphs AT martinazziluca anintroductiontotheregularitytheoryforellipticsystemsharmonicmapsandminimalgraphs |