Variational Methods in Shape Optimization Problems

Dorin Bucur, Giuseppe Buttazzo (auth.)9780817643591, 0817643591, 0817644032

The study of shape optimization problems encompasses a wide spectrum of academic research with numerous applications to the real world. In this work these problems are treated from both 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 and features:

* Presents foundational introduction to shape optimization theory

* Studies certain classical problems: the isoperimetric problem and the Newton problem involving the best aerodynamical shape, and optimization problems over classes of convex domains

* Treats optimal control problems under a general scheme, giving a topological framework, a survey of “gamma”-convergence, and problems governed by ODE

* Examines shape optimization problems with Dirichlet and Neumann conditions on the free boundary, along with the existence of classical solutions

* Studies optimization problems for obstacles and eigenvalues of elliptic operators

* Poses several open problems for further research

* Substantial bibliography and index

Driven by good examples and illustrations and requiring only a standard knowledge in the calculus of variations, differential equations, and functional analysis, the book can serve as a text for a graduate course in computational methods of optimal design and optimization, as well as an excellent reference for applied mathematicians addressing functional shape optimization problems.

Table of contents :
Introduction to Shape Optimization Theory and Some Classical Problems….Pages 1-29
Optimization Problems over Classes of Convex Domains….Pages 31-52
Optimal Control Problems: A General Scheme….Pages 53-74
Shape Optimization Problems with Dirichlet Condition on the Free Boundary….Pages 75-119
Existence of Classical Solutions….Pages 121-143
Optimization Problems for Functions of Eigenvalues….Pages 145-173
Shape Optimization Problems with Neumann Condition on the Free Boundary….Pages 175-203