Extremum problems for eigenvalues of elliptic operators:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2006
|
| Series: | Frontiers in mathematics
|
| Subjects: | |
| Online Access: | Beschreibung für Leser Inhaltsverzeichnis |
| Item Description: | Auch als Internetausgabe |
| Physical Description: | X, 202 S. graph. Darst. |
| ISBN: | 3764377054 9783764377052 |
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MARC
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| 100 | 1 | |a Henrot, Antoine |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Extremum problems for eigenvalues of elliptic operators |c Antoine Henrot |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2006 | |
| 300 | |a X, 202 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Frontiers in mathematics | |
| 500 | |a Auch als Internetausgabe | ||
| 650 | 4 | |a Maximums et minimums | |
| 650 | 4 | |a Opérateurs elliptiques | |
| 650 | 4 | |a Valeurs propres | |
| 650 | 4 | |a Eigenvalues | |
| 650 | 4 | |a Elliptic operators | |
| 650 | 4 | |a Maxima and minima | |
| 650 | 0 | 7 | |a Elliptischer Differentialoperator |0 (DE-588)4140057-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Eigenwert |0 (DE-588)4151200-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Extremalproblem |0 (DE-588)4439315-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Elliptischer Differentialoperator |0 (DE-588)4140057-4 |D s |
| 689 | 0 | 1 | |a Eigenwert |0 (DE-588)4151200-5 |D s |
| 689 | 0 | 2 | |a Extremalproblem |0 (DE-588)4439315-5 |D s |
| 689 | 0 | |5 DE-604 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-014970488 | ||
Record in the Search Index
| _version_ | 1804135617861255168 |
|---|---|
| adam_text | Contents
Preface
ix
1
Eigenvalues of elliptic operators
1
1.1
Notation and prerequisites
....................... 1
1.1.1
Notation and Sobolev spaces
................. 1
1.1.2
Partial differential equations
................. 2
1.2
Eigenvalues and eigenfunctions
.................... 4
1.2.1
Abstract spectral theory
.................... 4
1.2.2
Application to elliptic operators
............... 5
1.2.3
First Properties of eigenvalues
................ 8
1.2.4
Regularity of eigenfimctions
.................. 9
1.2.5
Some examples
......................... 9
1.2.6
Fredholm
alternative
...................... 11
1.3
Min-max
principles and applications
................. 12
1.3.1
Min-max
principles
....................... 12
1.3.2
Monotonicity
.......................... 13
1.3.3
Nodal domains
......................... 14
1.4
Perforated domains
........................... 15
2
Took
17
2.1 Schwarz
rearrangement
......................... 17
2.2 Steiner symmetrization......................... 18
2.2.1
Definition
............................ 18
2.2.2
Properties
............................ 20
2.2.3
Continuous
Steiner
symmetrization
.............. 21
2.3
Continuity of eigenvalues
....................... 23
2.3.1
Introduction
.......................... 23
2.3.2
Continuity with variable coefficients
............. 26
2.3.3
Continuity with variable domains (Dirichlet case)
...... 28
2.3.4
The case of Neumann eigenvalues
............... 33
2.4
Two general existence theorems
.................... 35
2.5
Derivatives of eigenvalues
....................... 37
Contente
2.5.1
Introduction
.......................... 37
2.5.2
Derivative
with respect to the domain
............ 38
2.5.3
Case of multiple eigenvalues
.................. 41
2.5.4
Derivative with respect to coefficients
............ 43
The first eigenvalue of the Laplacian-Dirichlet
45
3.1
Introduction
............................... 45
3.2
The Faber-Krahn inequality
...................... 45
3.3
The case of polygons
.......................... 46
3.3.1
An existence result
....................... 47
3.3.2
The cases
N = 3,4....................... 50
3.3.3
A challenging open problem.
.................. 51
3.4
Domains in a box
............................ 52
3.5
Multi-connected domains
....................... 55
The second eigenvalue of the Laplacian-Dirichlet
61
4.1
Minimizing
λ·2
.............................. 61
4.1.1
The Theorem of
Krahn-Szegö................. 61
4.1.2
Case of a connectedness constraint
.............. 63
4.2
A convexity constraint
......................... 63
4.2.1
Optimality conditions
..................... 64
4.2.2
Geometric properties of the optimal domain
......... 67
4.2.3
Another regularity result
................... 71
The other Dirichlet eigenvalues
73
5.1
Introduction
............................... 73
5.2
Connectedness of minimizers
..................... 74
5.3
Existence of a minimizer for
Аз
.................... 76
5.3.1
A concentration-compactness result
............. 76
5.3.2
Existence of a ininimizer
.................... 77
5.4
Case of higher eigenvalues
....................... 80
Functions of Dirichlet eigenvalues
85
6.1
Introduction
............................... 85
6.2
Ratio of eigenvalues
.......................... 86
6.2.1
The Ashbaugh-Benguria Theorem
.............. 86
6.2.2
Some other ratios
........................ 90
6.2.3
A collection of open problems
................. 92
6.3
Sums of eigenvalues
.......................... 93
6.3.1
Sums of eigenvalues
...................... 93
6.3.2
Sums of inverses
........................ 94
6.4
General functions of
λι
and
Аз
.................... 95
6.4.1
Description of the set
£ =
(Αι, Α2)
.............. 95
6.4.2
Existence of minimizers
.............
Q8
Contents
vii
7
Other boundary conditions for the Laplacian
101
7.1
Neumann boundary condition
..................... 101
7.1.1
Introduction
.......................... 101
7.1.2
Maximization of the second Neumann eigenvalue
...... 102
7.1.3
Some other problems
...................... 104
7.2
Robin boundary condition
....................... 106
7.2.1
Introduction
.......................... 106
7.2.2
The Bossel-Daners Theorem
.................. 107
7.2.3
Optimal insulation of conductors
............... 110
7.3
Stekloff eigenvalue problem
...................... 113
8
Eigenvalues of
Schrödinger
operators
117
8.1
Introduction
............................... 117
8.1.1
Notation
............................. 117
8.1.2
A general existence result
................... 119
8.2
Maximization or minimization of the first eigenvalue
........ 119
8.2.1
Introduction
.......................... 119
8.2.2
The maximization problem
.................. 119
8.2.3
The minimization problem
.................. 123
8.3
Maximization or minimization of other eigenvalues
......... 125
8.4
Maximization or minimization of the fundamental gap
À2
—
λι
. . 127
8.4.1
Introduction
.......................... 127
8.4.2
Single-well potentials
...................... 127
8.4.3
Minimization or maximization with an L00 constraint
. . . 131
8.4.4
Minimization or maximization with an Lp constraint
.... 134
8.5
Maximization of ratios
......................... 136
8.5.1
Introduction
.......................... 136
8.5.2
Maximization of
Аг(У)/Аі(У)
in one dimension
....... 136
8.5.3
Maximization of Xn(V)/Xi{V) in one dimension
...... 137
9
Non-homogeneous strings and membranes
141
9.1
Introduction
............................... 141
9.2
Existence results
............................ 143
9.2.1
A first general existence result
................ 143
9.2.2
A more precise existence result
................ 143
9.2.3
Nonlinear constraint
...................... 146
9.3
Minimizing or maximizing Xk(p) in dimension
1........... 148
9.3.1
Minimizing Xk(p)
........................ 149
9.3.2
Maximizing Xk(p)
....................... 150
9.4
Minimizing or maximizing Xk(p) in higher dimension
........ 152
9.4.1
Case of a ball
.......................... 152
9.4.2
General case
.......................... 153
9.4.3
Some extensions
........................ 155
viii
Contents
10
Optimal
conductivity
159
10.1
Introduction
............................... 159
10.2
The one-dimensional case
....................... 160
10.2.1
A general existence result
................... 160
10.2.2
Minimization or maximization of
Afe
(σ) ...........
161
10.2.3
Case of Neumann boundary conditions
............ 163
10.3
The general case
............................ 165
10.3.1
The maximization problem
.................. 165
10.3.2
The minimization problem
.................. 168
11
The bi-Laplacian operator
169
11.1
Introduction
............................... 169
11.2
The clamped plate
........................... 169
11.2.1
History
............................. 169
11.2.2
Notation and statement of the theorem
........... 170
11.2.3
Proof of the Rayleigh conjecture in dimension
N = 2,3 . . 171
11.3
Buckling of a plate
........................... 174
11.3.1
Introduction
.......................... 174
11.3.2
The case of a positive eigenfunction
............. 175
11.3.3
An existence result
....................... 177
11.3.4
The last step in the proof
................... 178
11.4
Some other problems
.......................... 181
11.4.1
Non-homogeneous rod and plate
............... 181
11.4.2
The optimal shape of a column
................ 183
References
187
Index
199
|
| adam_txt |
Contents
Preface
ix
1
Eigenvalues of elliptic operators
1
1.1
Notation and prerequisites
. 1
1.1.1
Notation and Sobolev spaces
. 1
1.1.2
Partial differential equations
. 2
1.2
Eigenvalues and eigenfunctions
. 4
1.2.1
Abstract spectral theory
. 4
1.2.2
Application to elliptic operators
. 5
1.2.3
First Properties of eigenvalues
. 8
1.2.4
Regularity of eigenfimctions
. 9
1.2.5
Some examples
. 9
1.2.6
Fredholm
alternative
. 11
1.3
Min-max
principles and applications
. 12
1.3.1
Min-max
principles
. 12
1.3.2
Monotonicity
. 13
1.3.3
Nodal domains
. 14
1.4
Perforated domains
. 15
2
Took
17
2.1 Schwarz
rearrangement
. 17
2.2 Steiner symmetrization. 18
2.2.1
Definition
. 18
2.2.2
Properties
. 20
2.2.3
Continuous
Steiner
symmetrization
. 21
2.3
Continuity of eigenvalues
. 23
2.3.1
Introduction
. 23
2.3.2
Continuity with variable coefficients
. 26
2.3.3
Continuity with variable domains (Dirichlet case)
. 28
2.3.4
The case of Neumann eigenvalues
. 33
2.4
Two general existence theorems
. 35
2.5
Derivatives of eigenvalues
. 37
Contente
2.5.1
Introduction
. 37
2.5.2
Derivative
with respect to the domain
. 38
2.5.3
Case of multiple eigenvalues
. 41
2.5.4
Derivative with respect to coefficients
. 43
The first eigenvalue of the Laplacian-Dirichlet
45
3.1
Introduction
. 45
3.2
The Faber-Krahn inequality
. 45
3.3
The case of polygons
. 46
3.3.1
An existence result
. 47
3.3.2
The cases
N = 3,4. 50
3.3.3
A challenging open problem.
. 51
3.4
Domains in a box
. 52
3.5
Multi-connected domains
. 55
The second eigenvalue of the Laplacian-Dirichlet
61
4.1
Minimizing
λ·2
. 61
4.1.1
The Theorem of
Krahn-Szegö. 61
4.1.2
Case of a connectedness constraint
. 63
4.2
A convexity constraint
. 63
4.2.1
Optimality conditions
. 64
4.2.2
Geometric properties of the optimal domain
. 67
4.2.3
Another regularity result
. 71
The other Dirichlet eigenvalues
73
5.1
Introduction
. 73
5.2
Connectedness of minimizers
. 74
5.3
Existence of a minimizer for
Аз
. 76
5.3.1
A concentration-compactness result
. 76
5.3.2
Existence of a ininimizer
. 77
5.4
Case of higher eigenvalues
. 80
Functions of Dirichlet eigenvalues
85
6.1
Introduction
. 85
6.2
Ratio of eigenvalues
. 86
6.2.1
The Ashbaugh-Benguria Theorem
. 86
6.2.2
Some other ratios
. 90
6.2.3
A collection of open problems
. 92
6.3
Sums of eigenvalues
. 93
6.3.1
Sums of eigenvalues
. 93
6.3.2
Sums of inverses
. 94
6.4
General functions of
λι
and
Аз
. 95
6.4.1
Description of the set
£ =
(Αι, Α2)
. 95
6.4.2
Existence of minimizers
.
Q8
Contents
vii
7
Other boundary conditions for the Laplacian
101
7.1
Neumann boundary condition
. 101
7.1.1
Introduction
. 101
7.1.2
Maximization of the second Neumann eigenvalue
. 102
7.1.3
Some other problems
. 104
7.2
Robin boundary condition
. 106
7.2.1
Introduction
. 106
7.2.2
The Bossel-Daners Theorem
. 107
7.2.3
Optimal insulation of conductors
. 110
7.3
Stekloff eigenvalue problem
. 113
8
Eigenvalues of
Schrödinger
operators
117
8.1
Introduction
. 117
8.1.1
Notation
. 117
8.1.2
A general existence result
. 119
8.2
Maximization or minimization of the first eigenvalue
. 119
8.2.1
Introduction
. 119
8.2.2
The maximization problem
. 119
8.2.3
The minimization problem
. 123
8.3
Maximization or minimization of other eigenvalues
. 125
8.4
Maximization or minimization of the fundamental gap
À2
—
λι
. . 127
8.4.1
Introduction
. 127
8.4.2
Single-well potentials
. 127
8.4.3
Minimization or maximization with an L00 constraint
. . . 131
8.4.4
Minimization or maximization with an Lp constraint
. 134
8.5
Maximization of ratios
. 136
8.5.1
Introduction
. 136
8.5.2
Maximization of
Аг(У)/Аі(У)
in one dimension
. 136
8.5.3
Maximization of Xn(V)/Xi{V) in one dimension
. 137
9
Non-homogeneous strings and membranes
141
9.1
Introduction
. 141
9.2
Existence results
. 143
9.2.1
A first general existence result
. 143
9.2.2
A more precise existence result
. 143
9.2.3
Nonlinear constraint
. 146
9.3
Minimizing or maximizing Xk(p) in dimension
1. 148
9.3.1
Minimizing Xk(p)
. 149
9.3.2
Maximizing Xk(p)
. 150
9.4
Minimizing or maximizing Xk(p) in higher dimension
. 152
9.4.1
Case of a ball
. 152
9.4.2
General case
. 153
9.4.3
Some extensions
. 155
viii
Contents
10
Optimal
conductivity
159
10.1
Introduction
. 159
10.2
The one-dimensional case
. 160
10.2.1
A general existence result
. 160
10.2.2
Minimization or maximization of
Afe
(σ) .
161
10.2.3
Case of Neumann boundary conditions
. 163
10.3
The general case
. 165
10.3.1
The maximization problem
. 165
10.3.2
The minimization problem
. 168
11
The bi-Laplacian operator
169
11.1
Introduction
. 169
11.2
The clamped plate
. 169
11.2.1
History
. 169
11.2.2
Notation and statement of the theorem
. 170
11.2.3
Proof of the Rayleigh conjecture in dimension
N = 2,3 . . 171
11.3
Buckling of a plate
. 174
11.3.1
Introduction
. 174
11.3.2
The case of a positive eigenfunction
. 175
11.3.3
An existence result
. 177
11.3.4
The last step in the proof
. 178
11.4
Some other problems
. 181
11.4.1
Non-homogeneous rod and plate
. 181
11.4.2
The optimal shape of a column
. 183
References
187
Index
199 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Henrot, Antoine |
| author_facet | Henrot, Antoine |
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| id | DE-604.BV021757345 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:33:55Z |
| indexdate | 2024-07-09T20:43:22Z |
| institution | BVB |
| isbn | 3764377054 9783764377052 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014970488 |
| oclc_num | 70167828 |
| open_access_boolean | |
| owner | DE-824 DE-355 DE-BY-UBR DE-634 DE-83 DE-11 DE-188 DE-20 DE-29T |
| owner_facet | DE-824 DE-355 DE-BY-UBR DE-634 DE-83 DE-11 DE-188 DE-20 DE-29T |
| physical | X, 202 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Frontiers in mathematics |
| spelling | Henrot, Antoine Verfasser aut Extremum problems for eigenvalues of elliptic operators Antoine Henrot Basel [u.a.] Birkhäuser 2006 X, 202 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Frontiers in mathematics Auch als Internetausgabe Maximums et minimums Opérateurs elliptiques Valeurs propres Eigenvalues Elliptic operators Maxima and minima Elliptischer Differentialoperator (DE-588)4140057-4 gnd rswk-swf Eigenwert (DE-588)4151200-5 gnd rswk-swf Extremalproblem (DE-588)4439315-5 gnd rswk-swf Elliptischer Differentialoperator (DE-588)4140057-4 s Eigenwert (DE-588)4151200-5 s Extremalproblem (DE-588)4439315-5 s DE-604 http://deposit.dnb.de/cgi-bin/dokserv?id=2797531&prov=M&dok_var=1&dok_ext=htm Beschreibung für Leser Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014970488&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Henrot, Antoine Extremum problems for eigenvalues of elliptic operators Maximums et minimums Opérateurs elliptiques Valeurs propres Eigenvalues Elliptic operators Maxima and minima Elliptischer Differentialoperator (DE-588)4140057-4 gnd Eigenwert (DE-588)4151200-5 gnd Extremalproblem (DE-588)4439315-5 gnd |
| subject_GND | (DE-588)4140057-4 (DE-588)4151200-5 (DE-588)4439315-5 |
| title | Extremum problems for eigenvalues of elliptic operators |
| title_auth | Extremum problems for eigenvalues of elliptic operators |
| title_exact_search | Extremum problems for eigenvalues of elliptic operators |
| title_exact_search_txtP | Extremum problems for eigenvalues of elliptic operators |
| title_full | Extremum problems for eigenvalues of elliptic operators Antoine Henrot |
| title_fullStr | Extremum problems for eigenvalues of elliptic operators Antoine Henrot |
| title_full_unstemmed | Extremum problems for eigenvalues of elliptic operators Antoine Henrot |
| title_short | Extremum problems for eigenvalues of elliptic operators |
| title_sort | extremum problems for eigenvalues of elliptic operators |
| topic | Maximums et minimums Opérateurs elliptiques Valeurs propres Eigenvalues Elliptic operators Maxima and minima Elliptischer Differentialoperator (DE-588)4140057-4 gnd Eigenwert (DE-588)4151200-5 gnd Extremalproblem (DE-588)4439315-5 gnd |
| topic_facet | Maximums et minimums Opérateurs elliptiques Valeurs propres Eigenvalues Elliptic operators Maxima and minima Elliptischer Differentialoperator Eigenwert Extremalproblem |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2797531&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014970488&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT henrotantoine extremumproblemsforeigenvaluesofellipticoperators |