Why quasi-open sets ?

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Table of contents

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
2.3. Capacity, quasi-open sets and quasi-continuous functions
2.4. Existence of optimal sets in a box
Chapter 3. Capacitary measures 
3.1. Sobolev spaces in Rd
3.2. Capacitary measures and the spaces H1 μ
3.3. Torsional rigidity and torsion function
3.4. PDEs involving capacitary measures
3.5. The γ-convergence of capacitary measures
3.6. The γ-convergence in a box of finite measure
3.7. Concentration-compactness principle for capacitary measures
Chapter 4. Subsolutions of shape functionals 
4.1. Introduction
4.2. Shape subsolutions for the Dirichlet Energy
4.3. Interaction between energy subsolutions
4.4. Subsolutions for spectral functionals with measure penalization
4.5. Subsolutions for functionals depending on potentials and weights
4.6. Subsolutions for spectral functionals with perimeter penalization
4.7. Subsolutions for spectral-energy functionals
Chapter 5. Shape supersolutions and quasi-minimizers 
5.1. Introduction and motivation
5.2. Preliminary results
5.3. Lipschitz continuity of energy quasi-minimizers
5.4. Shape quasi-minimizers for Dirichlet eigenvalues
5.5. Shape supersolutions of spectral functionals
5.6. Measurable sets of positive curvature
5.7. Subsolutions and supersolutions
Chapter 6. Spectral optimization problems in Rd 
6.1. Optimal sets for the kth eigenvalue of the Dirichlet Laplacian
6.2. Spectral optimization problems in a box revisited
6.3. Spectral optimization problems with internal constraint
6.4. Optimal sets for spectral functionals with perimeter constraint
6.5. Optimal potentials for Schr¨odinger operators
6.6. Optimal measures for spectral-torsion functionals
6.7. Multiphase spectral optimization problems
Chapter 7. Appendix: Shape optimization problems for graphs 
7.1. Sobolev space and Dirichlet Energy of a rectifiable set
7.2. Sobolev space and Dirichlet Energy of a metric graph
7.3. Some examples of optimal metric graphs
Bibliography