I have been working on a research problem of mine and came across the concept of Lipschitz domains. I am curious about whether it is possible to show that there always exists a bi-Lipschitz map from $\mathcal{X} \to \mathcal{Y}$ whenever $\mathcal{X}, \mathcal{Y} \subset \mathbb{R}^d$ are Lipschitz domains.
I came across a result which shows the converse, i.e, if there exists a bi-Lipschitz map between $\mathcal{X}$ and $\mathcal{Y}$ and $\mathcal{X}$ is a Lipschitz domain then $\mathcal{Y}$ is also a Lipschitz domain. From my understanding of the topic (which is relatively limited), I think that there should exist a map between Lipschitz domains. However, I haven't been able to prove this claim or find a reference that proves/disproves the claim.
Any leads/references will be greatly appreciated. Thanks!
With the definition you quoted, the existence of a bilipschitz homeomorphism is utterly wrong. The simplest example is $X={\mathbb R}$ and $Y=(0,1)\subset \mathbb R$. One can give more interesting examples when both domains are bounded, for instance, $X$ is the unit disk in $\mathbb R^2$ and $Y$ is an annulus in $\mathbb R^2$ bounded by two disjoint circles. In the latter case, $X, Y$ are not even homeomorphic.
On the other hand, if you assume that both $X, Y$ are relatively compact Lipschitz domains in $R^n$ and there exists a homeomorphism $\bar{X}\to \bar Y$, then there exists also a bilipschitz homeomorphism $\bar X\to \bar Y$, unless $n=5, n=4$, see
Tukia, P.; Väisälä, J., Lipschitz und quasiconformal approximation and extension, Ann. Acad. Sci. Fenn., Ser. A I, Math. 6, 303-342 (1981). ZBL0448.30021.
as well as
Luukkainen, Jouni, Lipschitz and quasiconformal approximation of homeomorphism pairs, Topology Appl. 109, No. 1, 1-40 (2001). ZBL0964.57023.