I’m reading Evan’s PDE and has some confusion about the global approximation in Sobolev space(bounded domain).
at Step 2 it construct $W_i\supset U_i$. My lecturer said the construction matters as we have space for the convolution of $\eta_{\epsilon_i}$ and $\zeta_iu$. However, we know that $f^\epsilon$ is defined on $U_\epsilon\subset U$, which means $f^\epsilon\to f$ interiorly. In that case, why we need space outside $U_i$ to make sense?
Here is the theorem and proof in Evans:
