The book discusses basic concepts of functional analysis, measure and integration theory, calculus of variations and duality and its applications to variational problems of non-convex nature, such as the Ginzburg-Landau system in superconductivity, shape optimization models, dual variational formulations for micro-magnetism and others. Numerical Methods for such and similar problems, such as models in flight mechanics and the Navier-Stokes system in fluid mechanics have been developed through the generalized method of lines, including their matrix finite dimensional approximations. It concludes with a review of recent research on Riemannian geometry applied to Quantum Mechanics and Relativity.
The book will be of interest to applied mathematicians and graduate students in applied mathematics. Physicists, engineers and researchers in related fields will also find the book useful in providing a mathematical background applicable to their respective professional areas.
Table of Contents
SECTION I: FUNCTIONAL ANALYSIS. Metric Spaces. Topological Vector Spaces. Hilbert Spaces. The Hahn-Banach Theorems and the Weak Topologies. Topics on Linear Operators. Spectral Analysis, a General Approach in Normed spaces. Basic Results on Measure and Integration. The Lebesgue Measure in Rn. Other Topics in Measure and Integration. Distributions. The Lebesgue and Sobolev Spaces. SECTION II: CALCULUS OF VARIATIONS, CONVEX ANALYSIS AND RESTRICTED OPTIMIZATION. Basic Topics on the Calculus of Variations. More topics on the Calculus of Variations. Convex Analysis and Duality Theory. Constrained Variational Optimization. On Central Fields in the Calculus of Variations. SECTION III: APPLICATIONS TO MODELS IN PHYSICS AND ENGINEERING. Global Existence Results and Duality for Non-Linear Models of Plates and Shells. A Primal Dual Formulation and a Multi-Duality Principle for a Non-Linear Model of Plates. On Duality Principles for One and Three-Dimensional Non-Linear Models in Elasticity. A Primal Dual Variational Formulation Suitable for a Large Class of Non-Convex Problems in Optimization. A Duality Principle and Concerning Computational Method for a Class of Optimal Design Problems in Elasticity. Existence and Duality Principles for the Ginzburg-Landau System in Superconductivity. Existence of Solution for an Optimal Control Problem Associated to the Ginzburg-Landau System in Superconductivity. Duality for a Semi-Linear Model in Micro-Magnetism. About Numerical Methods for Ordinary and Partial Differential Equations. On the Numerical Solution of First Order Ordinary Differential Equation Systems. On the Generalized Method of Lines and its Proximal Explicit and Hyper-Finite Difference Approaches. On the Generalized Method of Lines Applied to the Time Independent Incompressible Navier-Stokes System. A Numerical Method for an Inverse Optimization Problem through the Generalized Method of Lines. References.
Fabio Silva Botelho obtained a Ph.D in Mathematics from Virginia Tech, USA in 2009. Prior to that got his undergraduate and master degrees in Aeronautical Engineering from the Technological Institute of Aeronautics, ITA, SP, Brazil, in 1992 and 1996 respectively. From 2004 to 2015 he was an Assistant Professor at the Mathematics Department of Federal University of Pelotas in Brazil. From May 2015 to the present date, he has worked as an Adjunct Professor at the Department of Mathematics of Federal University of Santa Catarina, in Florianopolis, SC, Brazil. His field of research is Functional Analysis, Calculus of Variations, Duality and Numerical Analysis applied to problems in physics and engineering. He has published two books Functional Analysis and Applied Optimization in Banach Spaces (2014) and Real Analysis and Applications (2018), both with Springer. He is also the author of a generalization of the Method of Lines, a numerical method for solving partial differential equations in which the domain of the equation in question is discretized in lines and the concerning solution is written on these lines as functions of the boundary conditions and the domain boundary shape.