The course is devoted to the methods of classical variational calculus - functional extrema by Euler equations, second functional derivative, convexity or monotonicity. Further, it contains investigation of quadratic functional, generalized solution, Sobolev spaces and variational problem for elliptic PDE's.
1. Functional extremum, Euler equations.
2. Conditions for functional extremum.
3. Theorem on the minimum of a quadratic functional.
4. Construction of minimizing sequences and their convergence.
5. Choice of basis.
6. Sobolev spaces.
7. Traces. Weak formulation of the boundary conditions.
8. V-ellipticity. Lax-Milgram theorem.
9. Weak solution of boundary-value problems.
 S. V. Fomin, R. A. Silverman, Calculus of variations, Courier Dover Publications, Dover 2000
 K. Rektorys, Variational Methods In Mathematics, Science And Engineering, Springer, Berlin, 2001
 B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, London 2004
 B. Van Brunt, The calculus of variations, Birkhäuser, Basel 2004
 E. Giusti, Direct methods in the calculus of variations, World Scientific, Singapore 2003