Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. My main problem/question is:

(1) Show there exist a sequence of bi-Lipschitz (i.e injective Lipschitz function with Lipschitz inverse) maps $F_n : \overline{\Omega} \to F_n(\overline{\Omega})$ with the image $F_n(\overline{\Omega})$ compactly contained in $\Omega$, and such that $\lim_{n\to \infty} F_n(x) = x$ for all $x \in \overline{\Omega}$.

This will likely require the addition of assumptions on (the boundary of) $\Omega$.

I have a solution which involves adding the assumption that the boundary of $\Omega$ be the zero set of a $C^2$ (actually I think $C^{1,1}$ will suffice) function $G$ and letting $F_n$ be the flow (say to time $1/n$) of the ODE $X' = -\nabla G(X)$. This leads to a few more questions, such as;

(2) Has this been done before? (it must have been, so I guess I'm asking for a source)

(3) Is the existence of $G$ implied if I impose $\Omega$ to have a $C^2$ boundary? (I may ask this in a separate post)

[Edit: It has been pointed out that $\nabla G$ must be non-vanishing on $\partial\Omega$.]

I have a weaker proof for (1) which adds only the assumption that $\Omega$ be star-shaped (say at the origin). I then set $F_n(x) = (1-\frac{1}{n})x$.

Any and all other ideas, new ideas, references, modifications/improvements to mine, and improved generality in assumptions is appreciated, thank you all.

regularised distancemight be a good search term; see, for example, here for construction and basic properties, and here for further developments. $\endgroup$3more comments