Variational methods in shape optimization problems The study of shape optimization problems involves a wide area of academic research and applications to the real world. In this work these problems are treated from the classical and modern perspectives and target 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. Key topics: * Presents foundational introduction to shape optimization theory * Studies some classical problems: the isoperimetric problem and the Newton problem involving the best aerodynamical shape optimization problems over classes of convex domains * Treats optimal control problems under a general scheme giving a topological framework a survey of G convergence problems governed by ODE * Examines shape optimization problems with Dirichlet and Neumann condition on the free boundary the existence of classical solutions * Poses some open questions Driven by several good examples and illustrations the book requires only a standard knowledge in the calculus of variations differential equations and functional analysis. TOC:Preface * Introduction to shape optimization theory and some classical problems * Optimization problems over classes of convex domains * Optimal control problems: a general scheme * Shape optimization problems with Dirichlet condition on the free boundary * Existence of classical solutions * Optimization problems for functions of eigenvalues * Bibliography * Index