In Evans's PDE book, he proves that $C^{\infty}(\bar U)$ is dense in $W^{k,p}(U)$, provided that $\partial U$ is of class $C^1$. The characterization of $C^1$ boundary he uses is the following:

The domain $U\subset \Bbb R^n$ has $C^1$ boundary if for each $x_0\in\partial U$ there is an $r>0$ such that (relabeling the coordinate if necessary) $$ B_r(x_0)\cap U = \{ x\in B_r(x_0): x_n>\gamma(x_1,\dots,x_{n-1}) \} $$ for some $\gamma\in C^1(\Bbb R^{n-1})$.

The proof in the book uses this characterization, I don't have problem with the proof itself. However, there's another characterization that I usually see.

The domain $U\subset \Bbb R^n$ has $C^1$ boundary if for each $x_0\in\partial U$ there is an $r,r'>0$ and a $C^1$ diffeomorphism $\Phi$, sending $x_0$ to $0$, such that $$ \Phi(B_r(x_0)\cap U) \cap B_{r'}(0) = \{ y\in B_{r'}(0): y_n>0 \} $$

Needless to say, this half-ball is easier to work with. However, I don't know any result concerning the behaviour of $W^{k,p}(U)$ functions under composition with $\Phi^{-1}$. I don't know what would happen if I obtain an approximation $f\in C^{\infty}$ of $u\circ\Phi^{-1}$ in the $y$-coordinate. Would $f\circ\Phi$ approximate our original $u$ (at least locally in $B_r(x_0)$)?

I'd really appreciate if anyone could tell me the result in this direction, or give me some good references. Here is another approach with link to Evans' proof I referred to.

Edit: Perhaps I was not clear enough. I am aware that the 2 characterizations are equivalent. What I want to know is if there's a proof of this result using the second definition directly.

  • $\begingroup$ The two definitions are equivalent. Are you asking for a proof of density of smooth functions that directly uses the second definition? Why do you want to do this? $\endgroup$ – Jeff Nov 3 '17 at 21:27
  • $\begingroup$ @Jeff Precisely, that's what I want to know. As for why, there's no particular reason. $\endgroup$ – BigbearZzz Nov 3 '17 at 21:28
  • $\begingroup$ Perhaps a better reason could be that I what to understand the behaviour of $W^{k,p}$ under diffeomorphism. $\endgroup$ – BigbearZzz Nov 3 '17 at 21:30
  • $\begingroup$ Composition of $W^{k,p}$ functions with smooth functions is generally ok. You can easily check, just write down what the derivatives of the composition would be if everything were smooth, and then try to prove it using the definition of weak derivative (it should work out by making the change of variables given by the diffeomorphism in the definition of weak derivative). $\endgroup$ – Jeff Nov 3 '17 at 21:44
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    $\begingroup$ Probably not. Presumably you need $\Phi\in C^k$ to ensure the composition is in $W^{k,p}$. It should be simple to check by hand! $\endgroup$ – Jeff Nov 3 '17 at 21:55

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