L. C. Evans, Partial Differential Equations, Second Edition.
Graduate Studies in Mathematics, volume 19.
American Mathematical Society, Providence, RI, 2010.
Selected material from:
Chap. 2: Solution formulas for the transport and Laplace equations. (2.1 - 2.2)
Chap. 5: Basic theory of Sobolev spaces. (5.1 - 5.7)
Chap. 6: Applications of Sobolev spaces to elliptic PDEs. (6.1 - 6.2.2)
Additionally, the appendices in the book; they cover relevant background material from real analysis, functional analysis, and measure theory. Integration on boundaries, Stokes' theorem.
See also the teaching material for autumn 2016 and autumn 2014.