When looking at a (nonlinear degenerate) PDE like
one defines as weak solution as
Now I wonder about (2.1). The deduction of (2.2) is ok to me, as I can use the standard way with partial integration of the original PDE. But I can't seem to find the deduction / motivation of (2.1).
Note: I did omit a lot of details here, as I hope they are not neccessary (in the way, that the deduction is kind of standard / generalized). The screenshots are from "Homogenization of a nonlinear degenerate parabolic differential equation" by Cao & Yue (2013).