Faculdade de Ciências e Tecnologia

Introdução ao Cálculo das Variações

Code

10988

Academic unit

Faculdade de Ciências e Tecnologia

Department

Departamento de Matemática

Credits

6.0

Teacher in charge

José Maria Nunes de Almeida Gonçalves Gomes

Total hours

70

Teaching language

Português

Objectives

The student must obtain a basic knowledge of the following topics:

  • Indirect methods: The Euler Lagrange Equation and conditions for the existence of a minimizer of the Euler Lagrange Functional.
  • Direct methods: weak convergence, weak derivative, Sobolev Spaces, existence of minimizers for some sequentially weakly. semi-lower continuous functionals  in W1,2

Prerequisites

Basic knowledge in differential and integral Calculus and of Lp spaces Theory.

Subject matter

An Introduction to the Calculus of Variations

1. Indirect Methods


Classical Problems in the Calculus of Variations.

The first Variation

Euler-Lagrange Equations

Natural Boundary conditions.Convex functionals.

Constrained minimization problems (the Isoperimetric Problems)

Neumann boundary conditions

Inner Variation and Noether Theorem

The second variation (Legendre condition)

Examples. Applications.

 

2. Direct Methods

Main topics in Functional Analysis:  Ascoli-Arzela Theorem,  Baire''''s Lemma, Hahn Banach Theorem, Banach Steinhaus Theorem, Weak Topology, Convexity and Weak Topology,  Hilbert Spaces,  Sobolev Spaces W1,p.

Variational problems in W1,p. Tonelli''''s Theorem. Sufficient conditions for the existence of a minimizer to a variationa problem in W1,2. Existence of a classical solution inbetween well ordered sub and supersolutions (variationnal approach).

Applications.

Bibliography

Main references:

 One-dimensional Variational Problems (An introduction). Butazzo G., Giaquinta M. and Hildebrandt S., Oxford Science Publications.

Analyse Fonctionelle, Brézis H., Masson.

Measure Theory and Fine properties of functions, Evans L. and Gariepy R., CRC press series in advanced Mathematics.

Outros textos:

C. Fox — An Introduction to the Calculus of Variations, Dover, 1987

Gelfand, Fomin — Calculus of Variations, Dover, 2000

Sagan — Introduction to the Calculus of variations, McGraw-Hill, 1969

Teaching method

The Introductory course in Calculus of Variations consists in compreehensive sessions were both theoretical  lectures and Problem Solving sessions are included. Students must produce autonomous work and present resolutions in the blackboard. In the end of the semester, students are expected to study a scientific article on this subject.   

Evaluation method

Continuous evaluation including regular evaluation of individual homework. 

Courses