Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ and $W^{1,p}(\Omega')/W^{1,p}_0(\Omega')$ bi-Lipschitz equivalent? How much regularity do I need to assume for $\partial\Omega$ if this is false for general domains?
The two spaces are essentially function spaces on $\partial\Omega$, so identifying them is natural. I have understood that both spaces coincide with the Besov space $B^{1-1/p}_{p,p}(\partial\Omega)$ if $\partial\Omega$ is Lipschitz, the quotient norm being bi-Lipschitz to the Besov norm. However, I have never seen the bi-Lipschitz part explicitly stated, just equality of the spaces. Equality of spaces often implicitly means comparable norms, but I want to be sure that this is actually the case if there is no citeable or doable proof without using the boundary Besov space.
Somewhat related: Image of the trace operator.
To clarify, the main question is this: Are the quotient spaces bi-Lipschitz equivalent? If necessary, you can assume that the domain is Lipschitz, but I am also interested in the more general case. In the Lipschitz case the problem boils down to finding a reference to bi-Lipschitz equivalence to a Besov space.
