(Subject to revision according to progress in teaching plan).
L.C. Evans: Partial Differential Equations, 1998. Amer. Math. Soc.. ISBN: 0-8218-0772-2. Graduate Studies in Mathematics, Volume 19.
Material covered in Evans' book: Chapter 8 (Calculus of variations) except 8.3, 8.4.4 (b), Chapter 9 (Nonvariational techniques) except 9.4, 9.5, Chapters 11 + 3 (Conservation laws): 11.1, 11.4.2, 11.4.3 (including Theorem 3 but not the proof), 3.4.2, 3.4.3, 3.4.5.
Written lecture notes on weak convergence methods available here
Exercises (updated according to progress in teaching plan): Chapter 8 (Section 8.6): 2, 3, 4, 6, 7, 9, 10, 11, 12, 13. Chapter 9 (Section 9.7): 2, 4, 7, 8.