Consider the Sobolev embedding

**Thm 1:** Suppose , is a bounded domain in with boundary, , then

for any .

Consider the sharpness of Sobolev embedding, which means can not be bigger than .

WLOG assume . Choose a particular function , then when . The reason is the following

is smooth away from with

For any

Let , we have

Then

for any when . has weak derivative , only when .

Let us calculate

if and only if .

So if , we can find one such that . By the above analysis, such but .

**Thm 2:** Suppose , is a bounded domain in with boundary, , then the embedding is compact when .

Consider the sharpness of compact embedding, must be strictly less than . Actually we can find a sequence of but does not have convergent subsequence in .

As before assume . Choose , define , for , then

Since has compact support, then

Choose , then is bounded. However, has no convergent subsequence. Otherwise as , when , such subsequence must converge to 0 in . Since we have , apparently this can not be true.

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