Names of two manifold-like structures

Question

I'm looking for the canonical names of (and optionally a reference for) the following structures:

1. The result of cutting out, from a differentiable manifold homeomorphic to $\mathbb{R}^n$, an "open ball", i.e. an open, bounded region whose boundary is a closed, differentiable $n-1$ dimensional hypersurface
2. The same as 1., but let the closed hypersurface be non-differentiable somewhere.

Is the first perhaps a "solid $n$-torus"? And thus the second a "solid $n$-torus 'with corners' "?

Many thanks for any help/pointers!