Variational Methods in Shape Optimization Problems

1. Front Matter

   Pages i-viii

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2. Introduction to Shape Optimization Theory and Some Classical Problems

   Pages 1-29

3. Optimization Problems over Classes of Convex Domains

   Pages 31-52

4. Optimal Control Problems: A General Scheme

   Pages 53-74

5. Shape Optimization Problems with Dirichlet Condition on the Free Boundary

   Pages 75-119

6. Existence of Classical Solutions

   Pages 121-143

7. Optimization Problems for Functions of Eigenvalues

   Pages 145-173

8. Shape Optimization Problems with Neumann Condition on the Free Boundary

   Pages 175-203

9. Back Matter

   Pages 205-216

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The fascinating ?eld of shape optimization problems has received a lot of attention in recent years, particularly in relation to a number of applications in physics and engineering that require a focus on shapes instead of parameters or functions. The goal of these applications is to deform and modify the admissible shapes in order to comply with a given cost function that needs to be optimized. In this respect the problems are both classical (as the isoperimetric problem and the Newton problem of the ideal aerodynamical shape show) and modern (re?ecting the many results obtained in the last few decades). The intriguing feature is that the competing objects are shapes, i.e., domains of N R , instead of functions, as it usually occurs in problems of the calculus of va- ations. This constraint often produces additional dif?culties that lead to a lack of existence of a solution and to the introduction of suitable relaxed formulations of the problem. However, in certain limited cases an optimal solution exists, due to the special form of the cost functional and to the geometrical restrictions on the class of competing domains.

• Excel
• functional analysis
• optimization
• ordinary differential equation
• partial differential equation
• partial differential equations

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)

• Département de Mathématiques, Université de Metz, Metz Cedex 01, France

  Dorin Bucur

• Dipartimento di Matematica, Università di Pisa, Pisa, Italy

  Giuseppe Buttazzo

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