Functional Analysis and Applied Optimization in Banach Spaces: Applications to Non-Convex Variational Models

✍ Scribed by Fabio Botelho (auth.)

• Publisher: Springer International Publishing
• Year: 2014
• Tongue: English
• Leaves: 567
• Edition: 1
• Category: Library

✦ Synopsis

​This book introduces the basic concepts of real and functional analysis. It presents the fundamentals of the calculus of variations, convex analysis, duality, and optimization that are necessary to develop applications to physics and engineering problems. The book includes introductory and advanced concepts in measure and integration, as well as an introduction to Sobolev spaces. The problems presented are nonlinear, with non-convex variational formulation. Notably, the primal global minima may not be attained in some situations, in which cases the solution of the dual problem corresponds to an appropriate weak cluster point of minimizing sequences for the primal one. Indeed, the dual approach more readily facilitates numerical computations for some of the selected models. While intended primarily for applied mathematicians, the text will also be of interest to engineers, physicists, and other researchers in related fields.

✦ Table of Contents

Front Matter....Pages i-xviii
Front Matter....Pages 1-1
Topological Vector Spaces....Pages 3-40
The Hahn–Banach Theorems and Weak Topologies....Pages 41-56
Topics on Linear Operators....Pages 57-97
Basic Results on Measure and Integration....Pages 99-128
Other Topics in Measure and Integration....Pages 129-146
Distributions....Pages 147-166
The Lebesgue and Sobolev Spaces....Pages 167-174
Front Matter....Pages 175-221
Basic Concepts on the Calculus of Variations....Pages 223-223
Basic Concepts on Convex Analysis....Pages 225-249
Constrained Variational Optimization....Pages 251-285
Front Matter....Pages 287-317
Duality Applied to Elasticity....Pages 319-319
Duality Applied to a Plate Model....Pages 321-341
About Ginzburg–Landau-Type Equations: The Simpler Real Case....Pages 343-362
The Full Complex Ginzburg–Landau System....Pages 363-372
More on Duality and Computation for the Ginzburg–Landau System....Pages 373-392
On Duality Principles for Scalar and Vectorial Multi-well Variational Problems....Pages 393-424
More on Duality Principles for Multi-well Problems....Pages 425-446
Duality and Computation for Quantum Mechanics Models....Pages 447-464
Duality Applied to the Optimal Design in Elasticity....Pages 465-476
Front Matter....Pages 477-491
Duality Applied to Micro-Magnetism....Pages 319-319
The Generalized Method of Lines Applied to Fluid Mechanics....Pages 493-516
Duality Applied to the Optimal Control and Optimal Design of a Beam Model....Pages 517-535
Errata....Pages 537-552
Back Matter....Pages E1-E17
....Pages 553-560

✦ Subjects

Functional Analysis; Real Functions; Fourier Analysis; Numerical Analysis