(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 6 (Second order elliptic equations) -- Chapter 7 (Linear evolution equations) -- Chapter 5 and Appendices as needed by the above sections. -- Chapter 10 (Hamilton-Jacobi equations) -- If time permits, also parts of Chapter 11 (Systems of conservation law).