Calculus of variations and nonlinear partial differential equations: lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2008
|
| Series: | Lecture notes in mathematics
1927 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 196 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783540759133 3540759131 |
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MARC
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| 020 | |a 3540759131 |c Pb. : EUR 42.75 (freier Pr.), sfr 70.00 (freier Pr.) |9 3-540-75913-1 | ||
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| 245 | 1 | 0 | |a Calculus of variations and nonlinear partial differential equations |b lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 |c Luigi Ambrosio ... Ed.: Bernard Dacorogna ... |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2008 | |
| 300 | |a XI, 196 S. |b graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1927 | |
| 650 | 4 | |a Calcul des variations | |
| 650 | 4 | |a Équations aux dérivées partielles | |
| 650 | 4 | |a Calculus of variations |v Congresses | |
| 650 | 4 | |a Differential equations, Nonlinear |v Congresses | |
| 650 | 4 | |a Differential equations, Partial |v Congresses | |
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| 700 | 1 | |a Dacorogna, Bernard |d 1953- |e Sonstige |0 (DE-588)133791920 |4 oth | |
| 710 | 2 | |a Centro Internazionale Matematico Estivo |e Sonstige |0 (DE-588)1025933-8 |4 oth | |
| 830 | 0 | |a Lecture notes in mathematics |v 1927 |w (DE-604)BV000676446 |9 1927 | |
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Record in the Search Index
| _version_ | 1804137339214102528 |
|---|---|
| adam_text | Contents
Transport
Equation and Cauchy Problem for Non-Smooth
Vector Fields
Luigi Ambrosio
.................................................. 1
1
Introduction
.................................................. 1
2
Transport Equation and Continuity Equation
within the Cauchy-Lipschitz Framework
.......................... 4
3
ODE Uniqueness versus PDE Uniqueness
......................... 8
4
Vector Fields with a Sobolev Spatial Regularity
................... 19
5
Vector Fields with a BV Spatial Regularity
....................... 27
6
Applications
.................................................. 31
7
Open Problems, Bibliographical Notes, and References
............ 34
References
...................................................... 37
Issues in Homogenization for Problems
with
Non
Divergence Structure
Luis Caffarelli, Luis
Silvestre
...................................... 43
1
Introduction
.................................................. 43
2
Homogenization of a Free Boundary Problem: Capillary Drops
...... 44
2.1
Existence of a Minimizer
................................... 46
2.2
Positive Density Lemmas
................................... 47
2.3
Measure of the Free Boundary
.............................. 51
2.4
Limit as
ε
-> 0............................................ 53
2.5
Hysteresis
................................................ 54
2.6
References
................................................ 57
3
The Construction of Plane Like Solutions to Periodic Minimal
Surface Equations
............................................. 57
3.1
References
................................................ 64
4
Existence of Homogenization Limits for Fully Nonlinear Equations
... 65
4.1
Main Ideas of the Proof
.................................... 67
4.2
References
................................................ 73
References
...................................................... 74
X
Contents
A Visit with the oo-Laplace Equation
Michael G.
Granduli
.............................................. 75
1
Notation
..................................................... 78
2
The Lipschitz Extension/Variational Problem
..................... 79
2.1
Absolutely Minimizing Lipschitz iff Comparison With Cones
.... 83
2.2
Comparison With Cones Implies oo-Harmonic
................ 84
2.3
oo-Harmonic Implies Comparison with Cones
................. 86
2.4
Exercises and Examples
.................................... 86
3
From oo-Subharmonic to oo-Superharmonic
....................... 88
4
More Calculus of oo-Subharmonic Functions
...................... 89
5
Existence and Uniqueness
...................................... 97
6
The Gradient Flow and the Variational Problem
for IIIDuHU»
.................................................102
7
Linear on All Scales
...........................................105
7.1
Blow
Ups
and Blow Downs are Tight on a Line
...............105
7.2
Implications of Tight on a Line Segment
.....................107
8
An Impressionistic History Lesson
...............................109
8.1
The Beginning and
Gunnar Aronosson
.......................109
8.2
Enter Viscosity Solutions and R. Jensen
.....................
Ill
8.3
Regularity
................................................113
Modulus of Continuity
.....................................113
Harnack and Liouville
.....................................113
Comparison with Cones, Full Born
..........................114
Blowups are Linear
........................................115
Savin s Theorem
..........................................115
9
Generalizations, Variations, Recent Developments and Games
.......116
9.1
What is
Δοο
for H(x, u,
Du)!...............................116
9.2
Generalizing Comparison with Cones
........................118
9.3
The Metric Case
..........................................118
9.4
Playing Games
............................................119
9.5
Miscellany
................................................119
References
......................................................120
Weak
KAM
Theory and Partial Differential Equations
Lawrence C. Evans
...............................................123
1
Overview,
KAM
theory
........................................123
1.1
Classical Theory
..........................................123
The Lagrangian Viewpoint
.................................124
The Hamiltonian Viewpoint
.................................125
Canonical Changes of Variables, Generating Functions
.........126
Hamilton-Jacobi PDE
.....................................127
1.2 KAM
Theory
............................................127
Generating Functions, Linearization
..........................128
Fourier series
.............................................128
Small divisors
.............................................129
Contents
XI
Statement
of
KAM Theorem................................129
2
Weak
KAM
Theory: Lagrangian Methods
........................131
2.1
Minimizing Trajectories
....................................131
2.2 Lax-Oleinik
Semigroup
....................................131
2.3
The Weak
KAM
Theorem
..................................132
2.4
Domination
...............................................133
2.5
Flow
invariance,
characterization of the constant
с
............135
2.6
Time-reversal, Mather set
..................................137
3
Weak
KAM
Theory: Hamiltonian and PDE Methods
...............137
3.1
Hamilton-Jacobi PDE
.....................................137
3.2
Adding
Ρ
Dependence
.....................................138
3.3
Lions-Papanicolaou-
Varadhan
Theory
.......................139
A PDE construction of
Я
..................................139
Effective Lagrangian
.......................................140
Application: Homogenization of Nonlinear PDE
...............141
3.4
More PDE Methods
.......................................141
3.5
Estimates
................................................144
4
An Alternative Variational/PDE Construction
....................145
4.1
A new Variational Formulation
..............................145
A Minimax Formula
.......................................146
A New Variational Setting
..................................146
Passing to Limits
..........................................147
4.2
Application: Nonresonance and Averaging
....................148
Derivatives of Hfe
.........................................148
Nonresonance
.............................................148
5
Some Other Viewpoints and Open Questions
......................150
References
......................................................152
Geometrical Aspects of Symmetrization
Nicola
Fusco
....................................................155
1
Sets of finite perimeter
.........................................155
2 Steiner
Symmetrization of Sets of Finite Perimeter
................164
3
The
Polya-Szegö
Inequality
.....................................171
References
......................................................180
CIME
Courses on Partial Differential Equations and Calculus
of Variations
Elvira
Mascólo
...................................................183
|
| adam_txt |
Contents
Transport
Equation and Cauchy Problem for Non-Smooth
Vector Fields
Luigi Ambrosio
. 1
1
Introduction
. 1
2
Transport Equation and Continuity Equation
within the Cauchy-Lipschitz Framework
. 4
3
ODE Uniqueness versus PDE Uniqueness
. 8
4
Vector Fields with a Sobolev Spatial Regularity
. 19
5
Vector Fields with a BV Spatial Regularity
. 27
6
Applications
. 31
7
Open Problems, Bibliographical Notes, and References
. 34
References
. 37
Issues in Homogenization for Problems
with
Non
Divergence Structure
Luis Caffarelli, Luis
Silvestre
. 43
1
Introduction
. 43
2
Homogenization of a Free Boundary Problem: Capillary Drops
. 44
2.1
Existence of a Minimizer
. 46
2.2
Positive Density Lemmas
. 47
2.3
Measure of the Free Boundary
. 51
2.4
Limit as
ε
-> 0. 53
2.5
Hysteresis
. 54
2.6
References
. 57
3
The Construction of Plane Like Solutions to Periodic Minimal
Surface Equations
. 57
3.1
References
. 64
4
Existence of Homogenization Limits for Fully Nonlinear Equations
. 65
4.1
Main Ideas of the Proof
. 67
4.2
References
. 73
References
. 74
X
Contents
A Visit with the oo-Laplace Equation
Michael G.
Granduli
. 75
1
Notation
. 78
2
The Lipschitz Extension/Variational Problem
. 79
2.1
Absolutely Minimizing Lipschitz iff Comparison With Cones
. 83
2.2
Comparison With Cones Implies oo-Harmonic
. 84
2.3
oo-Harmonic Implies Comparison with Cones
. 86
2.4
Exercises and Examples
. 86
3
From oo-Subharmonic to oo-Superharmonic
. 88
4
More Calculus of oo-Subharmonic Functions
. 89
5
Existence and Uniqueness
. 97
6
The Gradient Flow and the Variational Problem
for IIIDuHU»
.102
7
Linear on All Scales
.105
7.1
Blow
Ups
and Blow Downs are Tight on a Line
.105
7.2
Implications of Tight on a Line Segment
.107
8
An Impressionistic History Lesson
.109
8.1
The Beginning and
Gunnar Aronosson
.109
8.2
Enter Viscosity Solutions and R. Jensen
.
Ill
8.3
Regularity
.113
Modulus of Continuity
.113
Harnack and Liouville
.113
Comparison with Cones, Full Born
.114
Blowups are Linear
.115
Savin's Theorem
.115
9
Generalizations, Variations, Recent Developments and Games
.116
9.1
What is
Δοο
for H(x, u,
Du)!.116
9.2
Generalizing Comparison with Cones
.118
9.3
The Metric Case
.118
9.4
Playing Games
.119
9.5
Miscellany
.119
References
.120
Weak
KAM
Theory and Partial Differential Equations
Lawrence C. Evans
.123
1
Overview,
KAM
theory
.123
1.1
Classical Theory
.123
The Lagrangian Viewpoint
.124
The Hamiltonian Viewpoint
.125
Canonical Changes of Variables, Generating Functions
.126
Hamilton-Jacobi PDE
.127
1.2 KAM
Theory
.127
Generating Functions, Linearization
.128
Fourier series
.128
Small divisors
.129
Contents
XI
Statement
of
KAM Theorem.129
2
Weak
KAM
Theory: Lagrangian Methods
.131
2.1
Minimizing Trajectories
.131
2.2 Lax-Oleinik
Semigroup
.131
2.3
The Weak
KAM
Theorem
.132
2.4
Domination
.133
2.5
Flow
invariance,
characterization of the constant
с
.135
2.6
Time-reversal, Mather set
.137
3
Weak
KAM
Theory: Hamiltonian and PDE Methods
.137
3.1
Hamilton-Jacobi PDE
.137
3.2
Adding
Ρ
Dependence
.138
3.3
Lions-Papanicolaou-
Varadhan
Theory
.139
A PDE construction of
Я
.139
Effective Lagrangian
.140
Application: Homogenization of Nonlinear PDE
.141
3.4
More PDE Methods
.141
3.5
Estimates
.144
4
An Alternative Variational/PDE Construction
.145
4.1
A new Variational Formulation
.145
A Minimax Formula
.146
A New Variational Setting
.146
Passing to Limits
.147
4.2
Application: Nonresonance and Averaging
.148
Derivatives of Hfe
.148
Nonresonance
.148
5
Some Other Viewpoints and Open Questions
.150
References
.152
Geometrical Aspects of Symmetrization
Nicola
Fusco
.155
1
Sets of finite perimeter
.155
2 Steiner
Symmetrization of Sets of Finite Perimeter
.164
3
The
Polya-Szegö
Inequality
.171
References
.180
CIME
Courses on Partial Differential Equations and Calculus
of Variations
Elvira
Mascólo
.183 |
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| ctrlnum | (OCoLC)181090520 (DE-599)DNB986835862 |
| dewey-full | 515.64 |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| genre | (DE-588)1071861417 Konferenzschrift 2005 Cetraro gnd-content |
| genre_facet | Konferenzschrift 2005 Cetraro |
| id | DE-604.BV023088001 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:39:58Z |
| indexdate | 2024-07-09T21:10:43Z |
| institution | BVB |
| institution_GND | (DE-588)1025933-8 |
| isbn | 9783540759133 3540759131 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016290908 |
| oclc_num | 181090520 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-824 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-83 DE-11 DE-188 DE-20 |
| owner_facet | DE-91G DE-BY-TUM DE-824 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-83 DE-11 DE-188 DE-20 |
| physical | XI, 196 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Calculus of variations and nonlinear partial differential equations lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 Luigi Ambrosio ... Ed.: Bernard Dacorogna ... Berlin [u.a.] Springer 2008 XI, 196 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1927 Calcul des variations Équations aux dérivées partielles Calculus of variations Congresses Differential equations, Nonlinear Congresses Differential equations, Partial Congresses Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2005 Cetraro gnd-content Variationsrechnung (DE-588)4062355-5 s Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s DE-604 Ambrosio, Luigi 1963- Sonstige (DE-588)133791408 oth Dacorogna, Bernard 1953- Sonstige (DE-588)133791920 oth Centro Internazionale Matematico Estivo Sonstige (DE-588)1025933-8 oth Lecture notes in mathematics 1927 (DE-604)BV000676446 1927 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016290908&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Calculus of variations and nonlinear partial differential equations lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 Lecture notes in mathematics Calcul des variations Équations aux dérivées partielles Calculus of variations Congresses Differential equations, Nonlinear Congresses Differential equations, Partial Congresses Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4128900-6 (DE-588)4062355-5 (DE-588)1071861417 |
| title | Calculus of variations and nonlinear partial differential equations lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 |
| title_auth | Calculus of variations and nonlinear partial differential equations lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 |
| title_exact_search | Calculus of variations and nonlinear partial differential equations lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 |
| title_exact_search_txtP | Calculus of variations and nonlinear partial differential equations lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 |
| title_full | Calculus of variations and nonlinear partial differential equations lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 Luigi Ambrosio ... Ed.: Bernard Dacorogna ... |
| title_fullStr | Calculus of variations and nonlinear partial differential equations lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 Luigi Ambrosio ... Ed.: Bernard Dacorogna ... |
| title_full_unstemmed | Calculus of variations and nonlinear partial differential equations lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 Luigi Ambrosio ... Ed.: Bernard Dacorogna ... |
| title_short | Calculus of variations and nonlinear partial differential equations |
| title_sort | calculus of variations and nonlinear partial differential equations lectures given at the c i m e summer school held in cetraro italy june 27 july 2 2005 |
| title_sub | lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005 |
| topic | Calcul des variations Équations aux dérivées partielles Calculus of variations Congresses Differential equations, Nonlinear Congresses Differential equations, Partial Congresses Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Calcul des variations Équations aux dérivées partielles Calculus of variations Congresses Differential equations, Nonlinear Congresses Differential equations, Partial Congresses Nichtlineare partielle Differentialgleichung Variationsrechnung Konferenzschrift 2005 Cetraro |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016290908&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
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