This book provides a self-contained presentation of the mathematical foundations, constructions, and tools necessary for studying problems where the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object. Shapes and Geometries: Analysis, Differential Calculus, and Optimization presents the extensive, recently developed theoretical foundation to shape optimization in a form that can be used by the engineering community. It also clearly explains the state-of-the-art developments in a mathematical language that will attract mathematicians to open questions in this important field.

Advanced engineers in various application areas use basic ideas of shape optimization, but often encounter difficulties due to the sophisticated mathematical foundations for the field. This new book challenges these difficulties by showing how the mathematics community has made extraordinary progress in establishing a rational foundation for shape optimization in past decades. This area of research has become very broad, rich, and fascinating from both theoretical and numerical standpoints. It is applicable in many different areas such as fluid mechanics, elasticity theory, modern theories of optimal design, free and moving boundary problems, optimal location and shape of geometric objects, and image processing.

The authors are among the most advanced mathematicians in the field of shape optimization. They are vastly experienced in fields of applications at significant levels of depth in both engineering and science. Their unique combination of mathematical and applications knowledge makes this book of great importance to both the mathematics and applications communities.