Winter quarter 2008

M W F 2:00 - 2:50 LCB 225

Instructor Andrej Cherkaev

Every problem of the calculus of variations has a solution, provided that the word `solution' is suitably understood. David Hilbert

Clear and elegant methods of modern Calculus of Variations allow to solve large number of problems in Science and Engineering. Originated by Bernoulli, Newton, Euler, and systematically developed beginning from XVIII century, these days Calculus of Variations attracts attention of mathematicians and provides new tools to find the best possible solutions, and to understand the essence of optimality.

Topics to be covered:

• Basic techniques of Calculus of Variations. Euler equations, Legendre and Weierstrass tests, direct methods.
• Hamiltonian.
• Duality and Legendre transform.
• Variational principles.
• Variational problems for multiple integrals:
• Optimization of shape of domains
• Variational problems for multiple integrals:
• Ill-posed problem. Introduction to relaxation. :

Addressed to graduate and to senior undergraduate students in math and science.

Notes:

1. Chaper 1. Introduction
2. Chapters 2 and 3 Stationarity. Development
3. Chapters 4 and 5 Inequality tests, Constrained problems. Introduction to Control theory
4. Chapter. Numerical methods. To be posted.
5. Chapters 6 and 7 Irregular solution, regularization and relaxation
6. Chapters 8 and 9 Multivariable Problems. Stationarity

Reading:

1. Robert Weinstock. Calculus of Variations with Applications to Physics and Engineering. Dover Publications, 1974.
2. I. M. Gelfand, S. V. Fomin Calculus of Variations Dover Publications, 2000