Variational Methods

Garant předmětu:

Lecturer:

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, 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:

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:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Matematické inženýrství (compulsory course of the specialization)
• Aplikovaná algebra a analýza (compulsory course in the program)
• Matematická fyzika (elective course)
• Matematické inženýrství (compulsory course in the program)
• Fyzikální elektronika - Počítačová fyzika (elective course)