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Functional Analysis and Applied Optimization in Banach Spaces

Applications to Non-Convex Variational Models

  • Book
  • © 2014

Overview

Authors:
  1. Fabio Botelho

    1. Department of Mathematics and Statistics, Federal University of Pelotas, Pelotas, Brazil

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  • Introduces the basic concepts of real and functional analysis
  • Contains introductory and advanced concepts in measure and integration
  • Applications will be of interest to applied mathematicians, physicists, and engineers
  • Includes supplementary material: sn.pub/extras

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Table of contents (24 chapters)

  1. Front Matter

    Pages i-xviii
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  2. Basic Functional Analysis

    1. Front Matter

      Pages 1-1
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    2. Topological Vector Spaces

      • Fabio Botelho
      Pages 3-40

    3. The Hahn–Banach Theorems and Weak Topologies

      • Fabio Botelho
      Pages 41-56

    4. Topics on Linear Operators

      • Fabio Botelho
      Pages 57-97

    5. Basic Results on Measure and Integration

      • Fabio Botelho
      Pages 99-128

    6. The Lebesgue Measure in \(\mathbb{R}^{n}\)

      • Fabio Botelho
      Pages 129-146

    7. Other Topics in Measure and Integration

      • Fabio Botelho
      Pages 147-166

    8. Distributions

      • Fabio Botelho
      Pages 167-174

    9. The Lebesgue and Sobolev Spaces

      • Fabio Botelho
      Pages 175-221

  3. Variational Convex Analysis

    1. Front Matter

      Pages 223-223
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    2. Basic Concepts on the Calculus of Variations

      • Fabio Botelho
      Pages 225-249

    3. Basic Concepts on Convex Analysis

      • Fabio Botelho
      Pages 251-285

    4. Constrained Variational Optimization

      • Fabio Botelho
      Pages 287-317

  4. Applications

    1. Front Matter

      Pages 319-319
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    2. Duality Applied to Elasticity

      • Fabio Botelho
      Pages 321-341

    3. Duality Applied to a Plate Model

      • Fabio Botelho
      Pages 343-362

    4. About Ginzburg–Landau-Type Equations: The Simpler Real Case

      • Fabio Botelho
      Pages 363-372

    5. The Full Complex Ginzburg–Landau System

      • Fabio Botelho
      Pages 373-392

    6. More on Duality and Computation for the Ginzburg–Landau System

      • Fabio Botelho
      Pages 393-424
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Keywords

About this book

​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.

Reviews

“The aim of the present book is to consider a variety of problems arising in applications in relation with non-convex variational models. … The book will be a valuable resource for students and researchers in applied mathematics, physics, mechanics, and engineering.” (Ján Lovíšek, Mathematical Reviews, August, 2015)

Authors and Affiliations

  • Department of Mathematics and Statistics, Federal University of Pelotas, Pelotas, Brazil

    Fabio Botelho

Bibliographic Information

  • Book Title: Functional Analysis and Applied Optimization in Banach Spaces

  • Book Subtitle: Applications to Non-Convex Variational Models

  • Authors: Fabio Botelho

  • DOI: https://doi.org/10.1007/978-3-319-06074-3

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2014

  • Hardcover ISBN: 978-3-319-06073-6Published: 03 July 2014

  • Softcover ISBN: 978-3-319-38206-7Published: 23 August 2016

  • eBook ISBN: 978-3-319-06074-3Published: 12 June 2014

  • Edition Number: 1

  • Number of Pages: XVIII, 560

  • Number of Illustrations: 6 b/w illustrations, 51 illustrations in colour

  • Topics: Functional Analysis, Real Functions, Fourier Analysis, Numerical Analysis

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