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Cover; Title Page; Copyright Page; Table of Contents; Preface; Résumé of the main results; Chapter 1 Introduction and Examples; 1.1. Shape optimization problems; 1.2. Why quasi-open sets?; 1.3. Compactness and monotonicity assumptions in the shape optimization; 1.4. Lipschitz regularity of the state functions; Chapter 2 Shape optimization problems in a box; 2.1. Sobolev spaces on metric measure spaces; 2.2. The strong-γ and weak-γ convergence of energy domains; 2.2.1. The weak-γ -convergence of energy sets; 2.2.2. The strong-γ -convergence of energy sets
2.2.3. From the weak-γ to the strong-γ -convergence2.2.4. Functionals on the class of energy sets; 2.3. Capacity, quasi-open sets and quasi-continuous functions; 2.3.1. Quasi-open sets and energy sets from a shape optimization point of view; 2.4. Existence of optimal sets in a box; 2.4.1. The Buttazzo-Dal Maso Theorem; 2.4.2. Optimal partition problems; 2.4.3. Spectral drop in an isolated box; 2.4.4. Optimal periodic sets in the Euclidean space; 2.4.5. Shape optimization problems on compact manifolds; 2.4.6. Shape optimization problems in Gaussian spaces
2.4.7. Shape optimization in Carnot-Caratheodory space2.4.8. Shape optimization in measure metric spaces; Chapter 3 Capacitary measures; 3.1. Sobolev spaces in Rd; 3.1.1. Concentration-compactness principle; 3.1.2. Capacity, quasi-open sets and quasi-continuous functions; 3.2. Capacitary measures and the spaces H1μ; 3.3. Torsional rigidity and torsion function; 3.4. PDEs involving capacitary measures; 3.4.1. Almost subharmonic functions; 3.4.2. Pointwise definition, semi-continuity and vanishing at infinity for solutions of elliptic PDEs
3.4.3. The set of finiteness Ωμ of a capacitary measure3.4.4. The resolvent associated to a capacitary measure μ; 3.4.5. Eigenvalues and eigenfunctions of the operator -Δ + μ; 3.4.6. Uniform approximation with solutions of boundary value problems; 3.5. The γ -convergence of capacitary measures; 3.5.1. Completeness of the γ -distance; 3.5.2. The γ -convergence of measures and the convergence of the resolvents Rμ; 3.7. Concentration-compactness principle for capacitary measures; 3.7.1. The γ -distance between comparable measures; 3.7.2. The concentration-compactness principle
Chapter 4 Subsolutions of shape functionals4.1. Introduction; 4.2. Shape subsolutions for the Dirichlet Energy; 4.3. Interaction between energy subsolutions; 4.3.1. Monotonicity theorems; 4.3.2. The monotonicity factors; 4.3.3. The two-phase monotonicity formula; 4.3.4. Multiphase monotonicity formula; 4.3.5. The common boundary of two subsolutions. Application of the two-phase monotonicity formula.; 4.3.6. Absence of triple points for energy subsolutions. Application of the multiphase monotonicity formula; 4.4. Subsolutions for spectral functionals with measure penalization
2.2.3. From the weak-γ to the strong-γ -convergence2.2.4. Functionals on the class of energy sets; 2.3. Capacity, quasi-open sets and quasi-continuous functions; 2.3.1. Quasi-open sets and energy sets from a shape optimization point of view; 2.4. Existence of optimal sets in a box; 2.4.1. The Buttazzo-Dal Maso Theorem; 2.4.2. Optimal partition problems; 2.4.3. Spectral drop in an isolated box; 2.4.4. Optimal periodic sets in the Euclidean space; 2.4.5. Shape optimization problems on compact manifolds; 2.4.6. Shape optimization problems in Gaussian spaces
2.4.7. Shape optimization in Carnot-Caratheodory space2.4.8. Shape optimization in measure metric spaces; Chapter 3 Capacitary measures; 3.1. Sobolev spaces in Rd; 3.1.1. Concentration-compactness principle; 3.1.2. Capacity, quasi-open sets and quasi-continuous functions; 3.2. Capacitary measures and the spaces H1μ; 3.3. Torsional rigidity and torsion function; 3.4. PDEs involving capacitary measures; 3.4.1. Almost subharmonic functions; 3.4.2. Pointwise definition, semi-continuity and vanishing at infinity for solutions of elliptic PDEs
3.4.3. The set of finiteness Ωμ of a capacitary measure3.4.4. The resolvent associated to a capacitary measure μ; 3.4.5. Eigenvalues and eigenfunctions of the operator -Δ + μ; 3.4.6. Uniform approximation with solutions of boundary value problems; 3.5. The γ -convergence of capacitary measures; 3.5.1. Completeness of the γ -distance; 3.5.2. The γ -convergence of measures and the convergence of the resolvents Rμ; 3.7. Concentration-compactness principle for capacitary measures; 3.7.1. The γ -distance between comparable measures; 3.7.2. The concentration-compactness principle
Chapter 4 Subsolutions of shape functionals4.1. Introduction; 4.2. Shape subsolutions for the Dirichlet Energy; 4.3. Interaction between energy subsolutions; 4.3.1. Monotonicity theorems; 4.3.2. The monotonicity factors; 4.3.3. The two-phase monotonicity formula; 4.3.4. Multiphase monotonicity formula; 4.3.5. The common boundary of two subsolutions. Application of the two-phase monotonicity formula.; 4.3.6. Absence of triple points for energy subsolutions. Application of the multiphase monotonicity formula; 4.4. Subsolutions for spectral functionals with measure penalization