Course syllabus

Tentative lecture plan

Lec 1-2. Introduction, convex function, Dirichlet's principle

Lec 3-6, One-dimensional problems

Lec 7-10, Sobolev spaces

Lec 11-15, Direct methods

Lec 16-17, time for repetition and extra problems

List of possible questions/subjects for the oral exam Download List of possible questions/subjects for the oral exam

Lecture notes

Slides from Lecture 1 Download Slides from Lecture 1

Notes from Lecture 1 Download Notes from Lecture 1

Notes from Lecture 2 Download Notes from Lecture 2

Notes for Lecture 3 Download Notes for Lecture 3

Some related references for Lecture 3: For some details on the Brachistochrone and the cycloid Links to an external site., see also page 45 in One-dimensional Variational Problems. Links to an external site. See also here Links to an external site. for two detailed proofs that the solution is a cycloid.

Notes for Lecture 4 Download Notes for Lecture 4

Notes for Lecture 5 Download Notes for Lecture 5

Notes for Lecture 6 Download Notes for Lecture 6

More notes for Lecture 6 Download More notes for Lecture 6

Further references for Sobolev spaces: Most of the needed results can be found in Juha Kinnunen's lecture notes Links to an external site. and/or Chapter 5 in Craig Evans PDE book Links to an external site..

Notes for Lecture 7 Download Notes for Lecture 7

Notes for Lecture 8 Download Notes for Lecture 8

Notes for Lecture 9 Download Notes for Lecture 9

Notes for Lecture 10 Download Notes for Lecture 10

Notes for Lecture 11 Download Notes for Lecture 11

Notes for Lecture 12 Download Notes for Lecture 12

Notes for Lecture 13 Download Notes for Lecture 13

Notes for Lecture 14 Download Notes for Lecture 14

Notes for Lecture 15 Download Notes for Lecture 15

Additional notes for Lecture 15 Download Additional notes for Lecture 15

Notes for Lecture 16 Download Notes for Lecture 16

 

For further reading on regularity of nonlinear PDEs: