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Variational Methods in Shape Optimization Problems : Progress in Nonlinear Differential Equations and Their Applications - Dorin Bucur

Variational Methods in Shape Optimization Problems

Progress in Nonlinear Differential Equations and Their Applications


Published: 1st July 2005
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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.

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)

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
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
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780817643591
ISBN-10: 0817643591
Series: Progress in Nonlinear Differential Equations and Their Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 216
Published: 1st July 2005
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.27
Weight (kg): 1.1