 Solution

Every boundary point of $D$ satisfies the exterior sphere condition. For a point on the inner spherical boundary, use a smaller ball inside the removed ball and tangent at that point; for a point on the cube, use a ball in a supporting exterior half-space. If the exterior ball has centre $y$ and radius $r$, a local positive harmonic barrier isfor $n\ge 3$, while in two dimensions use $w(x)=\log (|x-y|/r)$. Adding a sufficiently large positive multiple of a global superharmonic function extends the local barrier across the bounded domain. Hence every boundary point is regular by part (v), and the Perron method for the Dirichlet problem produces a harmonic function attaining the prescribed continuous boundary data. The maximum principle for harmonic functions gives uniqueness.