Code: 01VAM Variational Methods
Lecturer: prof. Dr. Ing. Michal Bene¹ Weekly load: 2 Assessment: ZK
Department: 14101 Credits: 3 Semester: W
Description:
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.
Contents:
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.
Recommended literature:
Key references:
[1] S. V. Fomin, R. A. Silverman, Calculus of variations, Courier Dover Publications, Dover 2000
[2] K. Rektorys, Variational Methods In Mathematics, Science And Engineering, Springer, Berlin, 2001

Recommended references:
[3] B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, London 2004
[4] B. Van Brunt, The calculus of variations, Birkhäuser, Basel 2004
[5] E. Giusti, Direct methods in the calculus of variations, World Scientific, Singapore 2003
Keywords:
Variational calculus, Gâteaux derivative, Fréchet derivative, functional extrema, convexity, monotonicity, quadratic functional, Sobolev spaces, weak solution, Lax-Milgram theorem.