Variational methods in shape optimization problems
著者
書誌事項
Variational methods in shape optimization problems
（Progress in nonlinear differential equations and their applications / editor, Haim Brezis, v. 65）
Birkhäuser, c2005
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注記
"This volume started as a collection of the lecture notes from two courses given in the academic year 2000-2001 by the authors at the Dipartimento di Matematica Universit di Pisa and at Scuola Normale Superiore di Pisa respectively."--Pref
Includes bibliographical references (p. [205]-213) and index
内容説明・目次
内容説明
Shape optimization problems are treated from the classical and modern perspectives
Targets 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
Requires only a standard knowledge in the calculus of variations, differential equations, and functional analysis
Driven by several good examples and illustrations
Poses some open questions.
目次
* 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: Sverak'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
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