Variational methods in some shape optimization problems (Publications of the Scuola Normale Superiore) - Softcover

Bucur, Dorin; Buttazzo, Giuseppe

Softcover

ISBN 10: 8876422978 ISBN 13: 9788876422973

Publisher: Edizioni della Normale, 2002

Synopsis

The study of shape optimization problems is a very wide field, both classical, as the isoperimetric problem and Newton problem of the best aerodynamical shape show, and modern, for all the recent results obtained in the last two, three decades. The fascinating feature is that the competing objects are shapes, i.e. domains of Rn, instead of functions, as usually occurs in problems of calculus of variations. This constraint often produces additional difficulties that lead to a lack of existence of a solution and the introduction of suitable relaxed formulations of the problem. However, in a few cases an optimal solution exists, due to the special form of the cost functional and to the geometrical restriction on the class of competing domains. This volume collects the lecture notes of two courses given in the academic year 2000/01 by the authors at the University of Pisa and at the Scuola Normale Superiore respectively. The courses were mainly addressed to Ph. D. students and required a background in the topics in functional analysis that are usually taught in undergraduate courses.

About this title

From the Back Cover

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.

Review

From the reviews:

"The book under review deals with some variational methods to treat shape optimization problems ... . The book contains a complete study of mathematical problems for scalar equations and eigenvalues, in particular regarding the existence of solutions in shape optimization. ... The main goal of the book is to focus on the existence of an optimal shape, necessary conditions of optimality, and stability of optimal solutions under some prescribed kind of perturbations." (Jan Sokolowski, Mathematical Reviews, Issue 2006 j)

“The authors predominantly analyze optimal shape and optimal control problems ... . The book, though slim, is rich in content and provides the reader with a wealth of information, numerous analysis and proof techniques, as well as useful references (197 items). ... Numerous nontrivial examples illustrate the theory and can please even those readers who are rather application-oriented.”­­­ (Jan Chleboun, Applications of Mathematics, Vol. 55 (5), 2010)

Edition details

• Publisher: Edizioni della Normale
• Publication date: 2002
• ISBN 10: 8876422978
• ISBN 13: 9788876422973
• Binding: Paperback
• Language: English
• Number of pages: 217