Bucur / Buttazzo Variational Methods in Shape Optimization Problems

2005
ISBN: 978-0-8176-4403-1
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 65, 217 Seiten, eBook

Reihe: Progress in Nonlinear Differential Equations and Their Applications

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.

Zielgruppe

Research

Autoren/Hrsg.

Bucur, Dorin

Buttazzo, Giuseppe

Inhaltsverzeichnis

* 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