Variational Methods
Code  Completion  Credits  Range  Language 

01VAM  ZK  3  2  Czech 
 Lecturer:
 Michal Beneš (guarantor)
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

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.
 Requirements:

Basic course of Calculus, Linear Algebra and Numerical Mathematics, variational methods (in the extent of the courses 01MA1, 01MAA24, 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. Vellipticity. LaxMilgram theorem.
9. Weak solution of boundaryvalue 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, CahnHilliard phasetransition theory etc.
 Study Objective:

Knowledge:
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.
Skills:
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
 Note:
 Timetable for winter semester 2019/2020:
 Timetable is not available yet
 Timetable for summer semester 2019/2020:
 Timetable is not available yet
 The course is a part of the following study plans:

 Matematické inženýrství (compulsory course of the specialization)
 Matematická fyzika (elective course)
 Informatická fyzika (elective course)