Delfour M.C., Zolesio J.-P. Shapes and Geometries: Analysis, Differential Calculus, and Optimization

Philadelphia: Society for Industrial and Applied Mathematics, 2001. — 482 pages.
ISBN: 0898714893.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.List of FiguresClassical Descriptions and Properties of Domains
Relaxation to Measurable Domains
Topologies Generated by Distance Functions
Oriented Distance Function and Smoothness of Sets
Optimization of Shape Functions
Transformations versus Flows of Velocities
Shape Derivatives and Calculus, and Tangential Differential Calculus
Shape Gradients under a State Equation Constraint
Elements of Bibliography
Index of Notation