Fabio Silva Botelho, PhD

Professor Adjunto, Departamento de Matemática

Universidade Federal de Santa Catarina - UFSC


Cálculo Avançado – 2019 -2 – Resultados da Primeira Avaliação


Cálculo Variacional


Análise Funcional – Curso de Verão – 2019 – Notas de Aula, Listas de Exercícios e resultados da segunda avaliação de 1/3/2019


Cálculo 1 – notas de aula


Livros:

Functional Analysis and Applied Optimization in Banach Spaces, Springer 2014.

Book review

Real Analysis and Applications, Springer 2018.


A Classical Description of Variational Quantum Mechanics and Related Models, Nova Science Publishing, 2017.

This work is dedicated in memoriam to David Bohm and Jiddu Krishnamurti


Pre-prints – Physics


1. arXiv:1812.04097 [pdf, ps, other]
A variational formulation for relativistic mechanics based on Riemannian geometry and its application to the quantum mechanics context
Fabio Botelho
Comments: 15 pages, new results based on the Weinberg approach for relativistic mechanics
Subjects: Analysis of PDEs (math.AP)

2. A Variational Formulation for the Relativistic Klein-Gordon Equation


3- arXiv:1908.04611  [pdf, ps, other

quant-ph

A variational formulation for relativistic mechanics, a new interpretation for the Bohr atomic model and some concerning applications

Authors: Fabio Botelho

Abstract: This article develops a variational formulation for the relativistic Klein-Gordon equation. The main results are obtained through an extension of the classical mechanics approach to a more general context, which in some sense, includes the quantum mechanics one. For the second part of the text, the definition of normal field and its relation with the wave function concept play a fundamental role in the main results establishment. Among the applications, we include a model with the presence of electromagnetic fields and also the modeling of a chemical reaction. Finally, in the last section, we present some results about the Spin operator in a relativistic context. △ Less

Submitted 16 October, 2019; v1 submitted 13 August, 2019; originally announced August 2019.

Comments: 38 pages, some minor mistakes and typos corrected

MSC Class: 81Q05


Pre-prints-Applied Mathematics


1-00 arXiv:1910.01118  [pdf, ps, other

math.OC

On duality principles for one and three-dimensional non-linear models in elasticity

Authors: Fabio Botelho

Abstract: In this article, we develop duality principles applicable to primal variational formulations found in the non-linear elasticity theory. As a first application, we establish the concerning results in details for one and three-dimensional models. We emphasize such duality principles are applicable to a larger class of variational optimization problems, such as non-linear models of plates and shells and other models in elasticity. Finally, we formally prove there is no duality gap between the primal and dual formulations, in a local extremal context. △ Less

Submitted 3 October, 2019; originally announced October 2019.

Comments: 13 pages. arXiv admin note: text overlap with arXiv:1712.02701

MSC Class: 49N15



1- arXiv:1909.01790  [pdf, ps, other

math.OC

A primal dual variational formulation suitable for a large class of non-convex problems in optimization

Authors: Fabio Botelho

Abstract: In this article we develop a new primal dual variational formulation suitable for a large class of non-convex problems in the calculus of variations. The results are obtained through basic tools of convex analysis, duality theory, the Legendre transform concept and the respective relations between the primal and dual variables. The novelty here is that the dual formulation is established also for the primal variables, however with a large domain region of concavity about a critical point. Finally, we formally prove there is no duality gap between the primal and dual formulations in a local extremal context. △ Less

Submitted 3 September, 2019; originally announced September 2019.

Comments: 8 pages

MSC Class: 49N15


Pre-print- On the solution of the Navier-Stokes system- announced, see it in the next line


arXiv:1908.09751  [pdf, ps, other

math.GM

On the generalized method lines applied to the time-independent incompressible Navier-Stokes system

Authors: Fabio Botelho

Abstract: In the first part of this article, we obtain a linear system whose the solution solves the time-independent incompressible Navier-Stokes system for the special case in which the external forces vector is a gradient. In a second step we develop approximate solutions, also for the time independent incompressible Navier-Stokes system, through the generalized method of lines. We recall that for such a method, the domain of the partial differential equation in question is discretized in lines and the concerning solution is written on these lines as functions of the boundary conditions and boundary shape. Finally, we emphasize these last main results are established through applications of the Banach fixed point theorem. △ Less

Submitted 11 August, 2019; originally announced August 2019.

Comments: 21 pages

MSC Class: 65N40; 65N06



0.1- arXiv:1907.02503  [pdf, ps, other

math.OC

A numerical method for an inverse optimization problem through the generalized method of lines

Authors: Fabio Silva Botelho

Abstract: This article develops a solution for an inverse problem through the generalized method of lines. We consider a Laplace equation on a domain with internal and external boundaries with standard Dirichlet boundary conditions. Also, we specify a third non-homogeneous Newmann type boundary condition for the external boundary, and consider the problem of finding the optimal shape for the internal boundary such that all the prescribed boundary conditions are satisfied. The novelty here presented is the application of the generalized method of lines as a tool to compute a solution for such an inverse optimization problem. △ Less

Submitted 4 July, 2019; originally announced July 2019.

Comments: 7 pages

MSC Class: 49N45; 65N40

1.0-0arXiv:1907.00200  [pdf, ps, other

math.OC

A duality principle and related numerical method for a class of shape optimization problems in elasticity

Authors: Fabio Botelho, Alexandre Molter

Abstract: In this article we develop a duality principle and concerning numerical method for a shape optimization problem in elasticity. We consider the problem of finding the optimal shape for an elastic solid which minimizes its structural inner energy resulting from the action of external loads to be specified. The main results are obtained through standard tools of convex analysis and duality theory. We emphasize our algorithm do not include a filter to process the results, so that the result obtained is indeed a critical point for the original optimization problem. Finally, in the last section, we present some numerical examples concerning applications of the theoretical results established. △ Less

Submitted 29 June, 2019; originally announced July 2019.

Comments: 7 pages

MSC Class: 49N15; 49Q10


1.0-arXiv:1906.07758  [pdf, ps, other

math.OC

Duality suitable for a class of non-convex optimization problems

Authors: Fabio Botelho

Abstract: In this article we develop a duality principle suitable for a large class of problems in optimization. The main result is obtained through basic tools of convex analysis and duality theory. We establish a correct relation between the critical points of the primal and dual formulations and formally prove there is no duality gap between such formulations, in a local extremal context.

Submitted 18 June, 2019; originally announced June 2019.

Comments: 7 pages

MSC Class: 49N15


1. arXiv:1904.12379  [pdf, ps, other

math.NA

On the generalized method of lines and its proximal explicit and hyper-finite difference approaches

Authors: Fabio Botelho

Abstract: This article firstly develops a proximal explicit approach for the generalized method of lines. In such a method, the domain of the PDE in question is discretized in lines and the equation solution is written on these lines as functions of the boundary conditions and domain shape. The main objective of introducing a proximal formulation is to minimize the solution error as a typical parameter… ▽ More

Submitted 7 May, 2019; v1 submitted 28 April, 2019; originally announced April 2019.

Comments: 18 pages, some typos corrected, a new result added

MSC Class: 65N40; 65N06


1.1. arXiv:1904.02286  [pdf, ps, other

math.OC

A primal dual variational formulation and a multi-duality principle for a non-linear model of plates

Authors: Fabio Botelho

Submitted 18 April, 2019; v1 submitted 3 April, 2019; originally announced April 2019.

Comments: 20 pages, some typos corrected, a new result added

MSC Class: 49N15; 74P99

2. arXiv:1903.06014 [pdf, ps, other]
A duality principle for non-convex optimization in Rn
Fabio Botelho
Comments: 13 pages, some typos and errors corrected, in this version all proof details have been provided
Subjects: Optimization and Control (math.OC)
3. arXiv:1902.08811 [pdf, ps, other]
A note on optimization in Rn
Fabio Botelho
Comments: 10 pages
Subjects: Optimization and Control (math.OC)
4. arXiv:1902.04448 [pdf, ps, other]
Existence of solution for an optimal control problem associated to the Ginzburg-Landau system in superconductivity
Fabio Botelho, Eduardo Pandini Barros
Comments: 8 pages
Subjects: Optimization and Control (math.OC)
5. arXiv:1812.04097 [pdf, ps, other]
A variational formulation for relativistic mechanics based on Riemannian geometry and its application to the quantum mechanics context
Fabio Botelho
Comments: 15 pages, new results based on the Weinberg approach for relativistic mechanics
Subjects: Analysis of PDEs (math.AP)
6. arXiv:1809.09575 [pdf, ps, other]
On central fields in the calculus of variations
Fabio Botelho
Comments: 12 pages, typos corrected
Subjects: Optimization and Control (math.OC)
7. arXiv:1804.06283 [pdf, ps, other]
On General Duality Principles for Non-Convex Variational Optimization with Applications to the Ginzburg-Landau System in Superconductivity
Fabio Botelho
Comments: 32 pages, typos corrected, other results added
Subjects: Optimization and Control (math.OC)
8. arXiv:1712.04809 [pdf, ps, other]
A duality principle for a semi-linear model in micro-magnetism
Fabio Botelho
Comments: 7 pages
Subjects: Optimization and Control (math.OC)
9. arXiv:1712.03552 [pdf, ps, other]
On the numerical solution of non-linear first order ordinary differential equation systems
Fabio Botelho
Comments: 9 pages
Subjects: Numerical Analysis (math.NA)
10. arXiv:1712.02701 [pdf, ps, other]
A duality principle for non-linear elasticity
Fabio Botelho
Comments: 10 pages, more typos and errors corrected, some parts of the text have been re-written
Subjects: Optimization and Control (math.OC)
11. arXiv:1712.01595 [pdf, ps, other]
Global existence results and duality for non-linear models of plates and shells
Fabio Botelho
Comments: 28 pages, some parts of the text have been re-written, variational nature of the dual formulations retaken
Subjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)

This work is dedicated in memoriam to Professor J.J. Telega

12. arXiv:1712.01031 [pdf, ps, other]
On duality principles for non-convex variational models applied to a Ginzburg-Landau type equation
Fabio Botelho
Comments: 14 pages, more typos corrected
Subjects: Optimization and Control (math.OC)

Artigos completos publicados em periódicos


1. Fabio Botelho, A note on Riemannian geometry and the relativistic quantum mechanics context. CIÊNCIA E NATURA, v. 40, p. 58, 2018.


2. Fabio Botelho.,A Variational Formulation for the Relativistic Klein-Gordon Equation. CIÊNCIA E NATURA, v. 40, p. 57, 2018.


3. Fabio Botelho, On the Generalized Method of Lines Applied to Ginzburg-Landau Type Equations. International Journal of Applied and Computational Mathematics, v. 1, p. pp 1-15, 2016.


4. Lucas dos Santos Fernandez, Alexandre Molter, Fabio Silva Botelho, Simultaneous topology optimization and proportional actuators localization. SeMA Journal, v. 23, p. --, 2016.


5. Fabio Botelho, On the Lagrange multiplier theorem in Banach spaces. Matemática Aplicada e Computacional (Cessou em 1997. Cont. ISSN 1807-0302 Computational & Applied Mathematics) , v. 32, p. 135-144, 2013.


6. Fabio Botelho. Existence of solution for the Ginzburg Landau system, a related optimal control problem and its computation by the generalized method of lines. Applied Mathematics and Computation , v. 218, p. 11976-11989, 2012.


7. Fabio Botelho, On duality principles for scalar and vectorial multi-well variational problems. Nonlinear Analysis , v. 75, p. 1904-1918, 2012.


8. Fabio Botelho, Dual Variational Formulations for a Non-linear Model of Plates. JOURNAL OF CONVEX ANALYSIS , v. 17, p. 131-158, 2010.