Variational Methods

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Code Completion Credits Range Language
01VAM ZK 3 2 Czech
Michal Beneš (guarantor)
Department of Mathematics

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.


Basic course of Calculus, Linear Algebra and Numerical Mathematics, variational methods (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA, 01NM, 01FA12 held at the FNSPE CTU in Prague).

Syllabus of lectures:

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.

Syllabus of tutorials:

Exercise makes part of the contents and is devoted to solution of particular examples in variational calculus - shortest path, minimal surface area, bending rod, Cahn-Hilliard phase-transition theory etc.

Study Objective:


Classical variational calculus - conditions for existence of functional extrema, Euler equations, extremum of quadratic functional, generalized solution of operator equation, Sobolev spaces and weak solution of boundary value problems for elliptic PDE.


Analysis of functional extrema, solution of common problems of variational calculus and determination of solution properties.

Study materials:

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

[6] B. S. Mordukhovich, Variational Analysis and Applications, Springer International Publishing, 2018

Time-table for winter semester 2019/2020:
Time-table is not available yet
Time-table for summer semester 2019/2020:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2019-10-18
For updated information see http://bilakniha.cvut.cz/en/predmet23926805.html