SF2723 Topics in Mathematics III: Variational Methods and Differential Equations

Informal course description: Variational techniques is one of the most powerful way to solve complicated differential equations, it is also the most beautiful. 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 matheamtics.

In this course we will restrict ourselves to applications of variational methods to differential equations. This is the most important application for scientific applications, but it will also serve as a good foundation for people that are interested in other applications such as optimization et.c. We will in particular aim to use variational methods to understand partial differential equations, but we will also discuss the ordinary differential equations of classical mechanics.

Course content:

• Euler-Lagrange and Hamilton-Jacobi equations of classical mechanics.

• Review of Hilbert and Banach spaces

• Calculus in Hilbert and Banach spaces

• Introduction to Sobolev spaces

• Direct methods in the calculus of variations

• Existence of solutions to partial differential equations of variational form

• Basic regularity theory and strong solutions for partial differential equations of variational form

• Some basic applications of variational methods to other areas of both pure and applied mathematics

 

Course Structure:

 

The course will consist of 15 lectures 2x45 minutes each.

Eligibility and prerequisites: General eligibility to the masters program. It will be very useful to know some basic Lebesgue theory and functional analysis, corresponding to SF2743 Advanced real analysis I.

Literature: Jürgen Jost and Xianquing Li-Jost ”Calculus of variations” Cambridge studies in advanced mathematics 64, Cambridge university press

Examination:

• Homework assignments

• Final written exam

Offered by: SCI/Mathematics

Examiner and contact person: John Andersson (examiner johnan@kth.se)