I would like to know why the last claim on the second page regarding the vanishing of the first term holds. Take $\lvert \alpha \rvert = 0$ for simplicity. If we write this term explicitly, we get the following:
$$ \lVert v^\varepsilon - u_\varepsilon \rVert_{L^p(V)}^p = \int_V \lvert (\eta_\varepsilon \ast u_\varepsilon)(x) - u_\varepsilon(x) \rvert^p dx = \int_{V + \lambda \varepsilon \textrm{e}_n} \lvert (\eta_\varepsilon \ast u)(y) - u(y) \rvert^p dy$$
I know that the mollifications converge in $L^p_\textrm{loc}$ but even though in the last integral the domain is compactly contained in $U$, it is not fixed and the domains get closer and closer to the boundary. How does one show that the integral converges to $0$ nevertheless?