  We prove that a hemisphere in the Euclidean space $R^{n+1}$, viewed as the
graph of a function, admits no smooth perturbations as graphs with mean
curvature $H\ge 1$ whose boundary equator is fixed up to $C^2$. This is an
extension of the \emph{Mean Curvature Rigidity} phenomenon discovered by Gromov
and Souam on non-compact totally umbilic hypersurfaces in space forms. The
proof uses a Tangency Principle. On the other hand, we show that there exist
nontrivial smooth perturbations with $H\ge 1$ on a great spherical cap whose
boundary is fixed up to $C^2$. Similar results hold true for perturbations
decreasing $H$, and for the $r$ mean curvature function $H_r$. This contrast
between rigidity and non-rigidity is even true in the 1-dimensional case for
circles and for discrete objects (polygons inscribed in a circle).
