(Subject to revision according to progress in teaching plan.)
L. C. Evans, Partial Differential Equations, Second Edition.
Graduate Studies in Mathematics, volume 19, American Mathematical Society, Providence, RI, 2010.
Chap. 8: Calculus of variations.
Chap. 9: Nonvariational techniques.
Chap. 3.4 & 11: Conservation laws.
Additionally, the appendices in the book; they cover relevant background material from real analysis, functional analysis, and measure theory.
Benoit Perthame, Kinetic formulation of conservation laws, Oxford University Press, 2002.
Chap. 2: Preliminary material (kinetic tools).
Chap. 3: Kinetic formulation of conservation laws.
Chap. 5: Regularizing effects (averaging lemmas).