
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.