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.

中文翻译：

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高维欧几里得球的体积衰减和浓度——偏微分方程和变分视角

这是一个众所周知的事实 - 可以通过基本微积分证明 - $\mathbb{R}^n$ 中单位球的体积衰减为零，同时集中在边界球附近的薄壳上作为 $n \nearrow \infty$。对于这一事实，从不同的角度提供了许多严格的证明和启发式论证，包括欧几里得几何、凸几何、巴拿赫空间理论、组合学、概率、离散几何等。 在这篇笔记中，我们通过椭圆的正则性理论给出了另外两个证明。偏微分方程和变分法。