Lecture notes on mean curvature flow, barriers and singular perturbations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Pisa
Ed. della Normale
2013
|
| Series: | Appunti / Scuola Normale Superiore
12 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XVIII, 329 S. graph. Darst. |
| ISBN: | 9788876424281 |
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MARC
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| 100 | 1 | |a Bellettini, Giovanni |d 1963- |e Verfasser |0 (DE-588)1049681592 |4 aut | |
| 245 | 1 | 0 | |a Lecture notes on mean curvature flow, barriers and singular perturbations |c Giovanni Bellettini |
| 264 | 1 | |a Pisa |b Ed. della Normale |c 2013 | |
| 300 | |a XVIII, 329 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Appunti / Scuola Normale Superiore |v 12 | |
| 650 | 4 | |a Curvature | |
| 650 | 4 | |a Flows (Differentiable dynamical systems) | |
| 650 | 0 | 7 | |a Krümmungsfluss |0 (DE-588)4618875-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Krümmungsfluss |0 (DE-588)4618875-7 |D s |
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| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-88-7642-429-8 |
| 810 | 2 | |a Scuola Normale Superiore |t Appunti |v 12 |w (DE-604)BV025597622 |9 12 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027194850&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027194850&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-027194850 | ||
Record in the Search Index
| _version_ | 1804152044827705344 |
|---|---|
| adam_text | Contents
Introduction
ix
1
Signed
distance
from a smooth boundary
1
1.1
Smoothness and eikonal equation
............. 4
1.2
Mean curvature vector and second fundamental form
. . 7
1.3
Expansion of the Hessian of the signed distance
..... 14
1.4
Tangential derivatives
................... 16
1.5
Appendix: second fundamental form using
pararnetriza-
tions
............................ 19
2
First variations
23
2.1
First variation of volume integrals
............ 24
2.2
First variation of the perimeter
.............. 25
2.3
Direction of maximal slope of the perimeter
....... 34
3
Smooth flows
37
3.1
Smooth mean curvature flows
............... 42
3.2
Examples
......................... 46
3.3
Extensions and tangential derivatives
........... 57
4
Huisken s
monotonicity
formula
59
4.1
The gradient estimate of
Ecker
and Huisken
....... 64
5
Inclusion principle
69
5.1
Appendix: on the maximum and minimum principles
. . 81
6
Extension of the evolution equation
to a neighbourhood
85
6.1
Appendix: parametric evolution equations
........ 95
6.1.1
Evolution of geometric quantities
........ 96
vi
(
¡iov,
mni
licllenitłi
7
Local well-posedness: the approach
of Evans and Spruck
103
7.1
Local existence
...................... 108
7.2
Uniqueness
........................ 115
7.3
Appendix
......................... 123
8
Grayson s example
127
8.1
The proof
......................... 129
9
De Giorgi
s
barriers
135
9.0.1
The minimal barrier
............... 141
9.1
Translation
invariance
................... 144
9.2
Examples
......................... 151
9.3
On the topological regularity of the minimal barrier
... 156
9.4
On the complement of the minimal barrier
........ 158
9.5
Appendix: abstract definition of barrier
......... 160
10
Inner and outer regularizations
10.1
On the evolution of an unbounded set
..........169
11
An example of fattening
11.1
Fattening of two tangent circles
.............. 176
11.2
An alternative proof
.................... 181
11.3
Notes
........................... 183
12
Ilmanen s interposition lemma
187
12.1
Cl
*
-regularity of dp
.................... 194
12.2
Appendix
......................... 203
13
The avoidance principle
207
13.1
Distance between the complement of two barriers
.... 209
13.2
Proof of Theorem
13.3.................. 211
14
Comparison between barriers and a generalized evolution
217
14.1
Comparison flows
..................... 217
14.2
Barriers and comparison flows
.............. 220
14.3
Barriers and level set evolution
.............. 224
14.4
Notes
........................... 227
15
One-dimensional analysis related
to a reaction-diffusion equation
229
15.1
The minimizer
γ
..................... 233
15.2
Linearization around
γ:
the operator
С
.......... 240
vii
Lecture Notes on Mean Curvature Flow. Barriers and Singular Perturbations
15.2.1
The operator
С
.................. 240
15.3
The shape function
.................... 244
16
Parabolic singular perturbations:
formal matched asymptotics
249
16.1
Formal series expansions
................. 253
16.1.1
Expansion of <p€
................. 253
16.1.2
Expansions of V€, ve, He and A€ i
....... 253
16.1.3
Expansion of s€
.................. 254
16.1.4
Initial interface
.................. 254
16.1.5
Extensions in
[O, T] x U
............. 255
16.2
Expansion far from the interface
............. 256
16.3
Expansion in a tubular neighbourhood of the interface
. 258
17
Parabolic singular perturbations:
convergence and error estimate
263
17.1
Statement of the main theorem
.............. 263
17.2
Definition of vf
...................... 267
17.3
Comparison with boundary and initial data
....... 271
17.4
Comparison
........................ 274
17.5
v~ is
a subsolution
.................... 276
17.5.1
Conclusion of the proof of Theorem
17.5 .... 282
17.6
Inclusion principle for smooth compact mean curvature
flows
............................ 284
17.7
Notes
........................... 286
17.7.1
Some references
................. 294
References
297
Index
317
Symbols
321
The aim of this book is to give an introduction
to mean curvature flow using, as much as possible,
a parametrization free approach. Some relevant aspects
of mean curvature flow are described, such as the role
of the signed distance function and the comparison
principle, and their use in the theory of barriers.
Some simple examples of singularities
are discussed. Finally, also making use of a formal
asymptotic inner and outer expansion, we prove
the convergence of the parabolic
Allen-Cahn s equation to mean curvature flow
for sufficiently short times, together with an error estimate.
|
| any_adam_object | 1 |
| author | Bellettini, Giovanni 1963- |
| author_GND | (DE-588)1049681592 |
| author_facet | Bellettini, Giovanni 1963- |
| author_role | aut |
| author_sort | Bellettini, Giovanni 1963- |
| author_variant | g b gb |
| building | Verbundindex |
| bvnumber | BV041748378 |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)884991011 (DE-599)BVBBV041748378 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041748378 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:04:28Z |
| institution | BVB |
| isbn | 9788876424281 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027194850 |
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| owner | DE-355 DE-BY-UBR DE-188 DE-11 DE-83 |
| owner_facet | DE-355 DE-BY-UBR DE-188 DE-11 DE-83 |
| physical | XVIII, 329 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Ed. della Normale |
| record_format | marc |
| series2 | Appunti / Scuola Normale Superiore |
| spelling | Bellettini, Giovanni 1963- Verfasser (DE-588)1049681592 aut Lecture notes on mean curvature flow, barriers and singular perturbations Giovanni Bellettini Pisa Ed. della Normale 2013 XVIII, 329 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Appunti / Scuola Normale Superiore 12 Curvature Flows (Differentiable dynamical systems) Krümmungsfluss (DE-588)4618875-7 gnd rswk-swf Krümmungsfluss (DE-588)4618875-7 s DE-604 Erscheint auch als Online-Ausgabe 978-88-7642-429-8 Scuola Normale Superiore Appunti 12 (DE-604)BV025597622 12 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027194850&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027194850&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Bellettini, Giovanni 1963- Lecture notes on mean curvature flow, barriers and singular perturbations Curvature Flows (Differentiable dynamical systems) Krümmungsfluss (DE-588)4618875-7 gnd |
| subject_GND | (DE-588)4618875-7 |
| title | Lecture notes on mean curvature flow, barriers and singular perturbations |
| title_auth | Lecture notes on mean curvature flow, barriers and singular perturbations |
| title_exact_search | Lecture notes on mean curvature flow, barriers and singular perturbations |
| title_full | Lecture notes on mean curvature flow, barriers and singular perturbations Giovanni Bellettini |
| title_fullStr | Lecture notes on mean curvature flow, barriers and singular perturbations Giovanni Bellettini |
| title_full_unstemmed | Lecture notes on mean curvature flow, barriers and singular perturbations Giovanni Bellettini |
| title_short | Lecture notes on mean curvature flow, barriers and singular perturbations |
| title_sort | lecture notes on mean curvature flow barriers and singular perturbations |
| topic | Curvature Flows (Differentiable dynamical systems) Krümmungsfluss (DE-588)4618875-7 gnd |
| topic_facet | Curvature Flows (Differentiable dynamical systems) Krümmungsfluss |
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