Introduction to Calculus of Variations
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An Introduction to the Calculus of Variations
1. Indirect Methods
Examples and Classical Problems in the Calculus of Variations.
The first Variation.
Natural Boundary conditions. Convex functionals.
Constrained minimization problems.
Neumann boundary conditions.
The second variation (Legendre condition).
Hamiltonian formulation and Hamilton Jacobi equation.
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 an ortogonal basis of eigenfuntions of the Laplace operator.
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