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I have two qeustions about Sobolev spaces:

Is there any Sobolev inequality that $Du$ bounded with $Lp$ norm $u$. For example $$||Du||_{Lp}\leq||u||_{Lq}$$ and no in $W^{1,p}$. And my second question What is the necessary condition for exist extention? I know the sufficient condition is that $\partial U $ must be $C^1$

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  1. It is not possible to bound the $L^q$ norm of $Du$ by the $L^q$ norm of $u$. The reason is that oscillations on small scale contribute much more to $Du$ than to $u$. A one-dimensional example: the functions $u_n= n^{-1}\sin nx$ have $L^q$ norm that tends to $0$, but $u_n'=\cos nx$ has $L^q$ norm bounded from below.

  2. A necessary condition for extension is known as measure density condition: there exists a constant $c>0$ such that $|U\cap B(x,r)|\ge c|B(x,r)|$ for all $x\in \Omega$ and all small radii $r$. For general $p$ this is a recent result of Hajlasz, Koskela and Tuominen (2008).

By the way, there are more general sufficient conditions for $U$ to be a Sobolev extension domain; for example, extension holds in domains with Lipschitz boundary, in uniform domains, locally uniform domains, etc. Here is a brief overview of the subject; note that it was written in 2002 and therefore is somewhat out of date.

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