The boundary of a domain whose interior is diffeomorphic to the ball We know that there are compact manifolds with diffeomorphic interiors but their boundaries are not homeomorphic (see the question Manifolds with homeomorphic interiors).
My question is about a very special case: 
Assume $D$ is a bounded open set in $R^n$ with smooth boundary.
If $D$ is diffeomorphic to the unit ball, is $\bar D$, the closure of $D$, diffeomorphic to the closed unit ball?
 A: First, your assumption imply that $\bar D$ is a compact smooth manifold with boundary a topological sphere (because is a simply connected homology sphere). So, $\bar D$ is a topological $n$-ball, by the generalized Schoenflies theorem. Now, this reduces to the smooth Schoenflies problem (still open in $\Bbb R^4$). Also, I don't know whether an exotic $(n-1)$-sphere can be smoothly embedded in $\Bbb R^n$.
A: The case $n=4$ is open as far as I know. 
The case $n=3$ follows since $\mathbb R^3$ is irreducible, so it contains no fake 3-disk, i.e. $\bar D$ must be the standard disk.
The case $n=5$ is equivalent to the smooth $4$-dimensional Poincare conjecture
(which is still open). Here is why: 


*

*Any homotopy $4$-sphere embeds smoothly into $\mathbb R^5$ (Sketch:  homology $4$-sphere bounds a contractible smooth manifold $C$ [Kervaire, "Smooth homology spheres and their fundamental groups", Theorem 3]. In our case $\partial C$ is simply-connected, so attaching a collar on the boundary one gets a contractible $5$-manifold that is simply-connected at infinity, and hence it is diffeomorphic to $\mathbb R^5$ by a result of Stallings).

*Any embedding of the standard $4$-sphere into $\mathbb R^5$ bounds a standard disk, see [Smale, "Differentiable and Combinatorial Structures on Manifolds", Corollary 1.3]. What Smale actually states is that any embedded $S^{n-1}$ in $\mathbb R^n$ bounds a standard disk unless $n=4$ or $7$. This was before he proved the h-cobodorsm theorem hence he excludes $7$.
Finally, as mentioned in comments if $n>5$, then $\bar D$ is diffeomorphic to the standard disk by the h-cobordism theorem (sketch: since by assumption $D$ is simply-connected at infinity, $\partial D$ is a homotopy sphere and $\bar D$ is a contractible smooth manifold, so removing a small ball in its interior results in h-cobordism between then standard sphere and the embedded one. Proving that this is an h-cobordism involves standard excision considerations in homology). 
