Extension of Sobolev functions

Question

Actually I am searching if $u$ belong to $W^{1,p}(\Omega)$ then why extension out side by zero does not belong to $W^{1,p}(\mathbb{R}^n)$ in general? Can some body help?

Answer 1

We can resume the two comment given by @DanielFischer and @GiuseppeNegro with the following result (see Leoni page 293):

If $u\in W^{1,p}(\Omega)$ with $p\in [1,\infty)$, then for almost every segment in $\Omega$ parallel to the coordinate axes, $u$ restricted to this segment is absolutely continuous. In fact, this result is part of a classification os Sobolev spaces and I suggest you to take a look in the book of Leoni that I have cited.

Also, it is worth to note that when the dimension is $1$, because of the last result, every Sobolev function has a continuous representative.