Problems linking the shape of a domain or the coefficients of an elliptic operator to the sequence of its eigenvalues are among the most fascinating of mathematical analysis. In this book, we focus on extremal problems. For instance, we look for a domain which minimizes or maximizes a given eigenvalue of the Laplace operator with various boundary conditions and various geometric constraints. We also consider the case of functions of eigenvalues. We investigate similar questions for other elliptic operators, such as the Schrödinger operator, non homogeneous membranes, or the bi-Laplacian, and we look at optimal composites and optimal insulation problems in terms of eigenvalues. Providing also a self-contained presentation of classical isoperimetric inequalities for eigenvalues and 30 open problems, this book will be useful for pure and applied mathematicians, particularly those interested in partial differential equations, the calculus of variations, differential geometry, or spectral theory.
Author(s): Antoine Henrot
Edition: 1
Year: 2006
Language: English
Pages: 205
Contents......Page 4
Preface......Page 8
1.1.1 Notation and Sobolev spaces......Page 10
1.1.2 Partial differential equations......Page 11
1.2.1 Abstract spectral theory......Page 13
1.2.2 Application to elliptic operators......Page 14
1.2.3 First Properties of eigenvalues......Page 17
1.2.5 Some examples......Page 18
1.2.6 Fredholm alternative......Page 20
1.3.1 Min-max principles......Page 21
1.3.2 Monotonicity......Page 22
1.3.3 Nodal domains......Page 23
1.4 Perforated domains......Page 24
2.1 Schwarz rearrangement......Page 26
2.2.1 Definition......Page 27
2.2.2 Properties......Page 29
2.2.3 Continuous Steiner symmetrization......Page 30
2.3.1 Introduction......Page 32
2.3.2 Continuity with variable coefficients......Page 35
2.3.3 Continuity with variable domains (Dirichlet case)......Page 37
2.3.4 The case of Neumann eigenvalues......Page 42
2.4 Two general existence theorems......Page 44
2.5.1 Introduction......Page 46
2.5.2 Derivative with respect to the domain......Page 47
2.5.3 Case of multiple eigenvalues......Page 50
2.5.4 Derivative with respect to coefficients......Page 52
3.2 The Faber-Krahn inequality......Page 54
3.3 The case of polygons......Page 55
3.3.1 An existence result......Page 56
3.3.2 The cases N = 3, 4......Page 59
3.3.3 A challenging open problem......Page 60
3.4 Domains in a box......Page 61
3.5 Multi-connected domains......Page 64
4.1.1 The Theorem of Krahn-Szegö......Page 70
4.2 A convexity constraint......Page 72
4.2.1 Optimality conditions......Page 73
4.2.2 Geometric properties of the optimal domain......Page 76
4.2.3 Another regularity result......Page 80
5.1 Introduction......Page 81
5.2 Connectedness of minimizers......Page 82
5.3.1 A concentration-compactness result......Page 84
5.3.2 Existence of a minimizer......Page 85
5.4 Case of higher eigenvalues......Page 88
6.1 Introduction......Page 92
6.2.1 The Ashbaugh-Benguria Theorem......Page 93
6.2.2 Some other ratios......Page 97
6.2.3 A collection of open problems......Page 99
6.3.1 Sums of eigenvalues......Page 100
6.3.2 Sums of inverses......Page 101
6.4.1 Description of the set ε = (λ[sub(1)], λ[sub(2)])......Page 102
6.4.2 Existence of minimizers......Page 105
7.1.1 Introduction......Page 108
7.1.2 Maximization of the second Neumann eigenvalue......Page 109
7.1.3 Some other problems......Page 111
7.2.1 Introduction......Page 113
7.2.2 The Bossel-Daners Theorem......Page 114
7.2.3 Optimal insulation of conductors......Page 117
7.3 Stekloff eigenvalue problem......Page 120
8.1.1 Notation......Page 123
8.2.2 The maximization problem......Page 125
8.2.3 The minimization problem......Page 129
8.3 Maximization or minimization of other eigenvalues......Page 131
8.4.2 Single-well potentials......Page 133
8.4.3 Minimization or maximization with an L[sup(∞)] constraint......Page 137
8.4.4 Minimization or maximization with an L[sup(p)] constraint......Page 140
8.5.2 Maximization of λ[sub(2)](V)/λ[sub(1)](V) in one dimension......Page 142
8.5.3 Maximization of λ[sub(n)](V )/λ[sub(1)](V) in one dimension......Page 143
9.1 Introduction......Page 146
9.2.2 A more precise existence result......Page 148
9.2.3 Nonlinear constraint......Page 151
9.3 Minimizing or maximizing λ[sub(k)](ρ) in dimension 1......Page 153
9.3.1 Minimizing λ[sub(k)](ρ)......Page 154
9.3.2 Maximizing λ[sub(k)](ρ)......Page 155
9.4.1 Case of a ball......Page 157
9.4.2 General case......Page 158
9.4.3 Some extensions......Page 160
10.1 Introduction......Page 163
10.2.1 A general existence result......Page 164
10.2.2 Minimization or maximization of λ[sub(k)](σ)......Page 165
10.2.3 Case of Neumann boundary conditions......Page 167
10.3.1 The maximization problem......Page 169
10.3.2 The minimization problem......Page 172
11.2.1 History......Page 173
11.2.2 Notation and statement of the theorem......Page 174
11.2.3 Proof of the Rayleigh conjecture in dimension N = 2, 3......Page 175
11.3.1 Introduction......Page 178
11.3.2 The case of a positive eigenfunction......Page 179
11.3.3 An existence result......Page 181
11.3.4 The last step in the proof......Page 182
11.4.1 Non-homogeneous rod and plate......Page 185
11.4.2 The optimal shape of a column......Page 187
References/Bibliography......Page 190
F......Page 203
S......Page 204
Z......Page 205
