Lawrence C. Evans: Partial Differential Equations, 1998. American Mathematical Society. ISBN: 0-8218-0772-2 (alk. paper). Ch.2 (p. 20-44), Ch. 4 (only the Fourier transform, p. 182-186), Ch. 5 (p. 239-284) except 5.8.2 b), 5.8.3 and 5.9.2 (that is p.279-281,285-289), Ch. 6 (p. 293-333) except 6.2.3 (that is p.302-307).
The appendices in Evans cover well the material from real analysis, functional analysis and measure theory which is needed for the course. These appendices are A.1. A.2, A.3, A.4, A.5, A.6, B.2 c e f h i, C.2, C.3 (just Theorem 4), C.4, C.7, D.1, D.2, D.3 a (just the part on linear functional analysis), D.3 b (just Riesz representasjon's theorem), E.1, E.2. E.3.