  It is a well-known fact -- which can be shown by elementary calculus -- that
the volume of the unit ball in $\mathbb{R}^n$ decays to zero and simultaneously
gets concentrated on the thin shell near the boundary sphere as $n \nearrow
\infty$. Many rigorous proofs and heuristic arguments are provided for this
fact from different viewpoints, including Euclidean geometry, convex geometry,
Banach space theory, combinatorics, probability, discrete geometry, etc. In
this note we give yet another two proofs via the regularity theory of elliptic
partial differential equations and calculus of variations.
