Shape optimization problems are treated from the classical and modern perspectives



Targets a broad audience of graduate students in pure and applied mathematics, as well as engineers requiring a solid mathematical basis for the solution of practical problems



Requires only a standard knowledge in the calculus of variations, differential equations, and functional analysis



Driven by several good examples and illustrations



Poses some open questions.

* 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: sverák'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 RN > Topological constraints for shape stability > The optimal cutting problem > Eigenvalues of the Neumann Laplacian * Bibliography * Index