Thread: Sobolev Spaces and some Mollifier stuff

1. Sobolev Spaces and some Mollifier stuff

Note that the proof in question is from book 'Partial Differential Equations by Lawrence Evans'.

Please view attachments for proof in question, basically I am trying to prove that for bounded $\displaystyle U$ any function $\displaystyle u \in W^{k,p}(U)$ can be approximated by smooth functions $\displaystyle u_{m} \in C^{\infty}\cap W^{k,p}(U)$.

The questions I have about the proof are the following:

It says choose $\displaystyle \epsilon_{i} > 0$ so small that $\displaystyle u^{i} := \eta_{\epsilon_{i}}\ast(\zeta_{i}u)$ satisfies (3) in attachment. Is the reason that both conditions of (3) are satisfied for small enough $\displaystyle \epsilon_{i}$ because of two basic results of mollifiers which says that for fixed i it follows that $\displaystyle u^{\epsilon_{i}} \rightarrow \zeta_{i}u$ as $\displaystyle \epsilon_{i} \rightarrow 0 \text{ a.e. } \text{ and if } \zeta_{i}u \in L^{P}(U) \text{ then } u^{\epsilon_{i}} \rightarrow \zeta_{i}u \text{ as }\epsilon_{i} \rightarrow 0 \text{ in } L^{P}(U)$?

Secondly, why is $\displaystyle W_{i}$ only defined for $\displaystyle i = 1, ...$ and not for $\displaystyle i =0$? Do we not need $\displaystyle \cup_{i=0}W$ to cover $\displaystyle U$ so that further in the proof together with a compactness argument we can show that there are finitely many non zero terms in the sum $\displaystyle v := \sum_{i=0}^{\infty}u^{i}$ as can be seen in attachment?

Thanks for any assistance, let me know if anything is unclear.