Let $U\_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U\_2$ be a neighborhood of $U\_1$. Does there exist a real-analytic diffeomorphism $\psi: U\_2 \to W\_2$ for some $W\_2\subset \mathbb R^3$ such that
$$ \psi(U\_1) \subset B \subset \psi(U\_2)$$
where $B$ is the unit ball centered at the origin?



Finding a real-analytic diffeomorphism
The answer is positive. First, the fact that $U\_1$ is simply connected implies that every connected component $S\_i$ ($1\le i\le n$) of its smooth boundary $\partial U\_1$ is diffeomorphic to the $2$-sphere (not completely obvious, but true). Second, after the Schönfliess theorem in dimension $3$, each of these $2$-spheres bounds a domain $D\_i$ smoothly ($C^\infty$) diffeomorphic with the compact $3$-ball. Since $U\_1$ is bounded and connected, it is easy to conclude that one of these ball domains, say $D\_1$, contains $U\_1$. So, $U\_1$ is the complement of $D\_2\cup\dots\cup D\_n$ in the interior of $D\_1$. Now, any smooth diffeomorphism $D\_1\to B$ is isotopic to the identity (by "Alexander's trick") and hence extends to a global smooth diffeomorphism $\phi:R^3\to R^3$. This provides a positive answer to your question but with instead of $\psi$ a diffeomorphism $(1-\epsilon)\phi$ which is only smooth. I guess that $U\_2$ is bounded. Hence, the Whitney approximation theorem gives on some neighborhood of $\bar U\_2$ a real analytic $\psi$ arbitrarily close to $(1-\epsilon)\phi$
in the $C^1$ topology (i.e. at every point $x\in\bar U\_2$, the value $\psi(x)$ and the three partial derivates $\partial \psi/\partial x(x)$, $\partial \psi/\partial y(x)$, $\partial \psi/\partial z(x)$ are close to those of $(1-\epsilon)\phi$.) If close enough, then $\psi$ still restricts to an embedding $U\_2\to R^3$ (as one sees using the local inversion theorem) and satisfies your demand.


