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
Lecture notes
Some related references for Lecture 3: For some details on the Brachistochrone and the cycloid, see also page 45 in One-dimensional Variational Problems. See also here for two detailed proofs that the solution is a cycloid.
Further references for Sobolev spaces: Most of the needed results can be found in Juha Kinnunen's lecture notes and/or Chapter 5 in Craig Evans PDE book.
Additional notes for Lecture 15
For further reading on regularity of nonlinear PDEs: