In my lecture notes I have most theorems about Sobolev spaces stated with Lipschitz boundaries, for instance
On the othe hand, the same theorems in Evans' PDE book are stated with $C^1$ boundaries. In fact, looks like Evans does not use Lipschitz boundaries at all as there is not even a definition of it.
So I wonder, does one imply the other? My guess is that $C^1$ boundaries are Lipschitz (and therefore Evans theorems are more general ?) because I found out on a quick google search that $C^1$ functions are Lipschitz, and the definitions of $C^1$ and Lipschitz boundaries involve a mapping of the same type respectively.
However the definition of Lipschitz boundary that I have coincides with wikipedia's definition of weakly Lipschitz boundary (I found that also surprising)
https://en.wikipedia.org/wiki/Lipschitz_domain#Generalization
seems a lot more complicated than that of $C^1$ boundary
, so I doubt the prove goes along the lines of just observing that $C^1$ functions are Lipschitz, as I initially thought before seeing the definitions. What do you think? Could you prove it (that $C^1$ boundaries are weakly Lipschitz I mean, using the definitions above )or tell me where to find a proof?
In $C^k$ boundary definition if we use $$l_p(y_1,...,y_n) = \frac{1}{r}\left(\left(y_1,...,y_{n-1},\frac{r(y_n-\gamma(y_1,...,y_{n-1}))}{\sup_y (y_n-\gamma(y_1,...,y_{n-1}))}\right)-(p_1,...,p_{n-1},0)\right)$$ then $$l_p(\partial \Omega \cap B_r(p)) = \{y:B_1(0): y_n = 0\}$$ $$l_p(\Omega \cap B_r(p)) = \{y:B_1(0): y_n > 0\}$$
Hence $\Omega$ is weakly lipschitz if $\gamma$ is $C^1$.