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Math Help - Trace Theorem

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    Trace Theorem

    I am currently working through notes on Sobolev Spaces in Partial Differential Equations by Lawrence.c.Evans

    Check 2 page attachment for relevant notes. It is the proof of the Trace Theorem.

    In the proof I would like to know why (1) is not trivial since \Gamma \subset \partial U and \partial U is a boundary which therefore has measure zero, does it not follow then that \int_{\Gamma} |u|^{p}dx' \leq \int_{\partial U }|u|^{p}dx' = 0, therefore \int_{\Gamma} |u|^{p}dx' = 0\text{ } \leq\text{ } C\int_{U}|u|^{p} + |Du|^{p}dx.

    And then would the last line ||u||_{L^{p}(\Gamma_{i})} \text{ } \leq \text{ } C||u||_{W^{1,p}(U)}   \text{  for    i=1,...,N} not also follow easily since 0 = ||u||_{L^{p}(\Gamma_{i})} \text{ } \leq \text{ } C||u||_{W^{1,p}(U)}   \text{   for   i=1,...,N}.

    Thanks
    Attached Thumbnails Attached Thumbnails Trace Theorem-attachment-1-definition.jpg   Trace Theorem-partial-differential-equations-lawrence.evans.jpg  
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