I am learning variational analysis from "Variational Analysis in Sobolev and BV Spaces - Applications to Pdes and Optimization" (https://epubs.siam.org/doi/book/10.1137/1.9781611973488) and i am not sure if i have found an error or there is something i am missing. In the proof of proposition 2.2.6 (pag. 24), where it is prooved that weak derivative, in the sense of distributions, coincides with classical derivative whenever classical derivative exists, the proof uses the integration by parts formula, a result that needs the open set $\Omega$ (which is the set where the functions we are working with are defined) to be bounded and have piecewise smooth boundary (https://en.wikipedia.org/wiki/Integration_by_parts#Theorem), but the book does not say anything about this, just uses the formula as if it es valid for an arbitrary open set $\Omega$. This same "error" (using integration by parts formula for arbitrary open $\Omega$) is repeated is pag.32, where he obtains the weak formulation for the Neumann boundary problem. So, i am wondering if is this an error and i should acept this results only in case $\Omega$ is bounded and have piecewise smooth boundary, or there is some way to overcome this dificult and apply integration by parts for an arbitrary open $\Omega$.
Thanks.