Integration by part questions

Question

I'm currently reading Evans' Partial Differential Equation, and I have some questions regarding Integration by parts.
So we need an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary to apply integration by part on a function $u\in C^1(\bar{U})$.
But I notice in definition of weak derivative, Evans makes no reference to the boundary of open sets. And also in a lot of proof, he makes no use of the boundary.
I want to ask that for weak derivatives, do we not care about whether the open set has $C^1$ boundary? Do we only care about if $\int_{U}uD^\alpha\phi dx = (-1)^{|\alpha|}\int_{U}D^\alpha u\phi dx$ ?

Answer 1

The definition of a weak derivative is that $D^\alpha u$ is a weak $\alpha$-derivative of $u$ if

$$\tag{1} \int_{U}uD^\alpha\phi dx = (-1)^{|\alpha|}\int_{U}D^\alpha u\phi dx$$

for all smooth functions $\varphi : \Omega \to \mathbb R$ with compact support, hence $\varphi$ is zero at the boundary $\partial \Omega$ of $\Omega$. In particular, if $u$ is smooth (so that the weak derivatives coincide with the classical one), then (1) holds for all such $\varphi$ and the boundary values of $u$ is irrelevant.