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 questionsDriven by several good examples and illustrations, the book requires only a standard knowledge in the calculus of variations, differential equations, and functional analysis.

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