Optimizing the shape of an object to make it the most efficient, resistant, streamlined, lightest, noiseless, stealthy or the cheapest is clearly a very old task. But the recent explosion of modeling and scientific computing has given this topic new life. Many new and interesting questions have been asked. A mathematical topic was born—shape optimization (or optimum design).

This book provides a self-contained introduction to modern mathematical approaches to shape optimization. The book assumes only an undergraduate-level understanding of the subject matter but tackles open questions in this vibrant field. The analytical and geometrical tools and methods for the study of shapes are developed. In particular, the text presents a systematic treatment of shape variations and optimization associated with the Laplace operator and the classical capacity. Emphasis is also put on differentiation with respect to domains and a FAQ on the usual topologies of domains is provided. The book ends with geometrical properties of optimal shapes, including the case where they do not exist.