Calculus of Variations

Kursstart: Fredag 19/1, kl 10

Course content

Variational techniques is one of the most powerful way to solve complicated differential equations. The area goes back to the Bernoullis, Newton, Leibniz and the history of variational techniques runns through Euler, Lagrange, Jacobi, Hilbert and many other of the most reknown mathematicians of forlorn days.

Calculus of variations is concerned with finding the minimal value of some function, in general a function from some infinite dimensional space to the real numbers. This appears to be a very narrow problem – but nothing could be further from the truth. Almost all of classical physics, and even much of modern physics, can be formulated in terms of variational problems. There are also many applications of variational techniques in pure mathematics.

In this course, we will study Calculus of Variations and its connection to differential equations. We will treat several classical problems that can be formulated in terms of calculus of variations and also discuss how to use variational methods to understand partial differential equations.

Content: (A more detailed but still preliminary plan can be found here)

Convex functions
The Dirichlet principle
Elliptic operators
The Brachistocrone problem
The Harmonic oscillator
Euler-Lagrange and Hamilton-Jacobi equations of classical mechanics
The Isoperimetric Inequality
Sobolev Spaces
The Direct Method in the Calculus of Variations
Regularity theory

Lecturer

Erik Lindgren, eriklin@kth.se

Course literature

Lecture notes: Handwritten lecture notes will be published online

Other relevant literature:

Examination

There will be up to four sets of homework and a final oral exam. The homework assignments will be posted in online, see assignments. Please mind the due date. We will not accept solutions handed in after the strict deadlines. Extra assignments to compensate unsatisfactory homework may to some extent be given at the end of the course.

During the oral exam, you will be asked about the homework problems and you will be offered theoretical questions. For instance, you may be asked to derive a formula or prove a theorem from the lectures. The complete list of such possible exam questions will be available in online at least three weeks before the exam.