Introduction to Calculus of Variations
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
Weekly hours
5
Total hours
70
Teaching language
Português
Subject matter
An Introduction to the Calculus of Variations
1. Indirect Methods
Examples and Classical Problems in the Calculus of Variations.
The first Variation.
Euler-Lagrange Equations.
Natural Boundary conditions. Convex functionals.
Constrained minimization problems.
Neumann boundary conditions.
The second variation (Legendre condition).
Hamiltonian formulation and Hamilton Jacobi equation.
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.