Preface |
Eigenvalues of elliptic operators / 1: |
Notation and prerequisites / 1.1: |
Notation and Sobolev spaces / 1.1.1: |
Partial differential equations / 1.1.2: |
Eigenvalues and eigenfunctions / 1.2: |
Abstract spectral theory / 1.2.1: |
Application to elliptic operators / 1.2.2: |
First Properties of eigenvalues / 1.2.3: |
Regularity of eigenfunctions / 1.2.4: |
Some examples / 1.2.5: |
Fredholm alternative / 1.2.6: |
Min-max principles and applications / 1.3: |
Min-max principles / 1.3.1: |
Monotonicity / 1.3.2: |
Nodal domains / 1.3.3: |
Perforated domains / 1.4: |
Tools / 2: |
Schwarz rearrangement / 2.1: |
Steiner symmetrization / 2.2: |
Definition / 2.2.1: |
Properties / 2.2.2: |
Continuous Steiner symmetrization / 2.2.3: |
Continuity of eigenvalues / 2.3: |
Introduction / 2.3.1: |
Continuity with variable coefficients / 2.3.2: |
Continuity with variable domains (Dirichlet case) / 2.3.3: |
The case of Neumann eigenvalues / 2.3.4: |
Two general existence theorems / 2.4: |
Derivatives of eigenvalues / 2.5: |
Derivative with respect to the domain / 2.5.1: |
Case of multiple eigenvalues / 2.5.3: |
Derivative with respect to coefficients / 2.5.4: |
The first eigenvalue of the Laplacian-Dirichlet / 3: |
The Faber-Krahn inequality / 3.1: |
The case of polygons / 3.3: |
An existence result / 3.3.1: |
The cases N = 3,4 / 3.3.2: |
A challenging open problem / 3.3.3: |
Domains in a box / 3.4: |
Multi-connected domains / 3.5: |
The second eigenvalue of the Laplacian-Dirichlet / 4: |
Minimizing λ2 / 4.1: |
The Theorem of Krahn-Szegö / 4.1.1: |
Case of a connectedness constraint / 4.1.2: |
A convexity constraint / 4.2: |
Optimality conditions / 4.2.1: |
Geometric properties of the optimal domain / 4.2.2: |
Another regularity result / 4.2.3: |
The other Dirichlet eigenvalues / 5: |
Connectedness of minimizers / 5.1: |
Existence of a minimizer for λ3 / 5.3: |
A concentration-compactness result / 5.3.1: |
Existence of a minimizer / 5.3.2: |
Case of higher eigenvalues / 5.4: |
Functions of Dirichlet eigenvalues / 6: |
Ratio of eigenvalues / 6.1: |
The Ashbaugh-Benguria Theorem / 6.2.1: |
Some other ratios / 6.2.2: |
A collection of open problems / 6.2.3: |
Sums of eigenvalues / 6.3: |
Sums of inverses / 6.3.1: |
General functions of λ1 and λ2 / 6.4: |
Description of the set ϵ = (λ1, λ2) / 6.4.1: |
Existence of minimizers / 6.4.2: |
Other boundary conditions for the Laplacian / 7: |
Neumann boundary condition / 7.1: |
Maximization of the second Neumann eigenvalue / 7.1.1: |
Some other problems / 7.1.3: |
Robin boundary condition / 7.2: |
The Bossel-Daners Theorem / 7.2.1: |
Optimal insulation of conductors / 7.2.3: |
Stekloff eigenvalue problem / 7.3: |
Eigenvalues of Schrödinger operators / 8: |
Notation / 8.1: |
A general existence result / 8.1.2: |
Maximization or minimization of the first eigenvalue / 8.2: |
The maximization problem / 8.2.1: |
The minimization problem / 8.2.3: |
Maximization or minimization of other eigenvalues / 8.3: |
Maximization or minimization of the fundamental gap λ2 - λ1 / 8.4: |
Single-well potentials / 8.4.1: |
Minimization or maximization with an L∞ constraint / 8.4.3: |
Minimization or maximization with an Lp constraint / 8.4.4: |
Maximization of ratios / 8.5: |
Maximization of λ2(V)/λ1(V) in one dimension / 8.5.1: |
Maximization of λn(V)/λ1(V) in one dimension / 8.5.3: |
Non-homogeneous strings and membranes / 9: |
Existence results / 9.1: |
A first general existence result / 9.2.1: |
A more precise existence result / 9.2.2: |
Nonlinear constraint / 9.2.3: |
Minimizing or maximizing λk(ρ) in dimension 1 / 9.3: |
Minimizing λk(ρ) / 9.3.1: |
Maximizing λk(ρ) / 9.3.2: |
Minimizing or maximizing λk(ρ) in higher dimension / 9.4: |
Case of a ball / 9.4.1: |
General case / 9.4.2: |
Some extensions / 9.4.3: |
Optimal conductivity / 10: |
The one-dimensional case / 10.1: |
Minimization or maximization of λk(σ) / 10.2.1: |
Case of Neumann boundary conditions / 10.2.3: |
The general case / 10.3: |
The bi-Laplacian operator / 10.3.1: |
The clamped plate / 11.1: |
History / 11.2.1: |
Notation and statement of the theorem / 11.2.2: |
Proof of the Rayleigh conjecture in dimension N = 2, 3 / 11.2.3: |
Buckling of a plate / 11.3: |
The case of a positive eigenfunction / 11.3.1: |
The last step in the proof / 11.3.3: |
Non-homogeneous rod and plate / 11.4: |
The optimal shape of a column / 11.4.2: |
References |
Index |
Preface |
Eigenvalues of elliptic operators / 1: |
Notation and prerequisites / 1.1: |
Notation and Sobolev spaces / 1.1.1: |
Partial differential equations / 1.1.2: |
Eigenvalues and eigenfunctions / 1.2: |