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Math Help - Sobolev Spaces and some Mollifier stuff

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    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 U any function u \in W^{k,p}(U) can be approximated by smooth functions u_{m} \in C^{\infty}\cap W^{k,p}(U).

    The questions I have about the proof are the following:

    It says choose \epsilon_{i} > 0 so small that 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 \epsilon_{i} because of two basic results of mollifiers which says that for fixed i it follows that u^{\epsilon_{i}} \rightarrow \zeta_{i}u as \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 W_{i} only defined for i = 1, ... and not for i =0? Do we not need \cup_{i=0}W to cover 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 v := \sum_{i=0}^{\infty}u^{i} as can be seen in attachment?

    Thanks for any assistance, let me know if anything is unclear.
    Attached Thumbnails Attached Thumbnails Sobolev Spaces and some Mollifier stuff-proof-attachment-1.jpg   Sobolev Spaces and some Mollifier stuff-proof-attachment-2.jpg  
    Last edited by Johnyboy; September 25th 2013 at 06:31 AM.
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