Extension of diffeomorphism up to boundary

Question

Suppose that I have two bounded open sets with $C^2$ boundary in $\mathbb{R}^n$ and I have a differmorphism $f:D_1\to D_2$ between them. I'm looking for some examples where the map $f$ can not be extend $C^2$ (I guess not even homeomorphism) up to boundary. Namely, their boundary, as embedded $C^2$ submanifold in $\mathbb{R}^n$ are not diffeomorphic under the extension.Any help will be appreciated!

Answer 1

Let $D_1 = D_2 = \{ x^2 + y^2 <1\}$ and let

$$ f : D_1 \to D_2, \ \ f(r, \theta) = (r, g(r) + \theta)$$

where $g : [0,1)\to \mathbb R$ is a smooth function so that $g(x) = 0$ on $[0,\epsilon)$ and $g(r) \to \infty$ as $r\to 1$. This $f$ cannot be extended continuously to the boundary.