Antoine Henrot, Michel Pierre 'Shape Variation and Optimization, A Geometrical Analysis' EMS, 2018, 379 pp.

摘要

Shapes are appealing objects to many people. Athletes try to get themselves into optimal shape for a sports event, but in this book shape refers to geometric two- or three-dimensional objects. One of the oldest shape optimization problems is the isoperimetric inequality: Given the length of a curve that bounds a plane set, what is the shape of the largest enclosed set? There are numerous proofs that the disc maximizes area; the shortest one that I could think of is in [3]. Other classical problems ask for the shape of cylindrical bars that maximize torsional rigidity, given that they consist of certain proportions of two materials. A hollow pipe consisting of metal and air provides a good example, and nature tells us from looking at a straw that this may be an optimal shape. Analysis and differentiation are essential tools in finding extrema of functionals. That is why one expects variational methods as appropriate ingredients in finding optimal shapes. To make the admissible geometric objects amenable to some sort of calculus of variation one needs an analytical description of their geometry. The book tries to bridge this gap and shows how fruitful the marriage of analysis and geometry can be. It is based on the authors' French version [1] from 2005, with suitable additions and updates.