Variational Methods in Shape Optimization Problems

Preface

* Introduction to Shape Optimization Theory and Some Classical Problems

> General formulation of a shape optimization problem

> The isoperimetric problem and some of its variants

> The Newton problem of minimal aerodynamical resistance

> Optimal interfaces between two media

> The optimal shape of a thin insulating layer

* Optimization Problems Over Classes of Convex Domains

> A general existence result for variational integrals

> Some necessary conditions of optimality

> Optimization for boundary integrals

> Problems governed by PDE of higher order

* Optimal Control Problems: A General Scheme

> A topological framework for general optimization problems

> A quick survey on "gamma"-convergence theory

> The topology of "gamma"-convergence for control variables

> A general definition of relaxed controls

> Optimal control problems governed by ODE

> Examples of relaxed shape optimization problems

* Shape Optimization Problems with Dirichlet Condition on the Free Boundary

> A short survey on capacities

> Nonexistence of optimal solutions

> The relaxed form of a Dirichlet problem

> Necessary conditions of optimality

> Boundary variation

> Continuity under geometric constraints

> Continuity under topological constraints: Šverák’s result

> Nonlinear operators: necessary and sufficient conditions for the "gamma"-convergence

> Stability in the sense of Keldysh

> Further remarks and generalizations

* Existence of Classical Solutions

> Existence of optimal domains under geometrical constraints

> A general abstract result for monotone costs

> The weak "gamma"-convergence for quasi-open domains

> Examples of monotone costs

> The problem of optimal partitions

> Optimal obstacles

* Optimization Problems for Functions of Eigenvalues

> Stability of eigenvalues under geometric domain perturbation

> Setting the optimization problem

> A short survey on continuous Steiner symmetrization

> The case of the first two eigenvalues of the Laplace operator

> Unbounded design regions

> Some open questions

* Shape Optimization Problems with Neumann Condition on the Free Boundary

> Some examples

> Boundary variation for Neumann problems

> General facts in R^N

> Topological constraints for shape stability

> The optimal cutting problem

> Eigenvalues of the Neumann Laplacian

* Bibliography

* Index