Variational Methods in Shape Optimization Problems

ISBN-13: 9780817643591 / Angielski / Twarda / 2005 / 216 str.

Dorin Bucur; Giuseppe Buttazzo

Opis

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.

Recenzje

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)

Spis treści

* Preface * Introduction to Shape Optimization Theory and Some Classical Problems > General formulation of a shape optimization problem > The isoperimetric problem and some of its variants > The Newton problem of minimal aerodynamical resistance > Optimal interfaces between two media > The optimal shape of a thin insulating layer * Optimization Problems Over Classes of Convex Domains > A general existence result for variational integrals > Some necessary conditions of optimality > Optimization for boundary integrals > Problems governed by PDE of higher order * Optimal Control Problems: A General Scheme > A topological framework for general optimization problems > A quick survey on 'gamma'-convergence theory > The topology of 'gamma'-convergence for control variables > A general definition of relaxed controls > Optimal control problems governed by ODE > Examples of relaxed shape optimization problems * Shape Optimization Problems with Dirichlet Condition on the Free Boundary > A short survey on capacities > Nonexistence of optimal solutions > The relaxed form of a Dirichlet problem > Necessary conditions of optimality > Boundary variation > Continuity under geometric constraints > Continuity under topological constraints: Šverák’s result > Nonlinear operators: necessary and sufficient conditions for the 'gamma'-convergence > Stability in the sense of Keldysh > Further remarks and generalizations * Existence of Classical Solutions > Existence of optimal domains under geometrical constraints > A general abstract result for monotone costs > The weak'gamma'-convergence for quasi-open domains > Examples of monotone costs > The problem of optimal partitions > Optimal obstacles * Optimization Problems for Functions of Eigenvalues > Stability of eigenvalues under geometric domain perturbation > Setting the optimization problem > A short survey on continuous Steiner symmetrization > The case of the first two eigenvalues of the Laplace operator > Unbounded design regions > Some open questions * Shape Optimization Problems with Neumann Condition on the Free Boundary > Some examples > Boundary variation for Neumann problems > General facts in RN > Topological constraints for shape stability > The optimal cutting problem > Eigenvalues of the Neumann Laplacian * Bibliography * Index

O autorze

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. 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.

Blurb

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.