(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), Chapter 9 (Nonvariational techniques, except 9.2.2, 9.3, 9.4, 9.5) , Chapter 10 (Hamilton-Jacobi equations, except 10.3.4).