Optimal quantitative stability estimates for Alexandrov's Soap Bubble Theorem via Gagliardo-Nirenberg-type interpolation inequalities

TL;DR

This paper establishes optimal quantitative stability estimates for Alexandrov's Soap Bubble Theorem in smooth domains, revealing new stability profiles for certain curvature deviations and demonstrating how regularity enhances these estimates.

Contribution

It introduces new stability profiles for curvature deviations in Alexandrov's Soap Bubble Theorem, especially for r ≤ (N-1)/2, and shows how higher regularity improves these estimates.

Findings

Stability estimates are optimal within C^{k,α} domains.

New stability profiles for r ≤ (N-1)/2 are established.

Higher regularity (larger k) improves the stability profile, approaching linearity.

Abstract

The paper provides optimal quantitative stability estimates for the celebrated Alexandrov's Soap Bubble Theorem within the class of $C^{k,\alpha}$ domains, for any $k \ge 1$ and $0 < \alpha \leq 1$, by leveraging Gagliardo-Nirenberg-type interpolation inequalities. Optimal estimates of uniform closeness to a ball are established for $L^r$ deviations of the mean curvature from being constant, for any $r\geq 2$ (more generally, for any $r>1$ such that $r\geq (2N-2)/(N+1)$). For $r>\frac{N-1}{2}$, the stability profile is linear, thus returning the existing results established in the literature through computations for nearly spherical sets. All the stability estimates for $r\le \frac{N-1}{2}$, for which the profile is not linear, are new; even in the particular case $r=2$ (which has been extensively studied, since it is a case of interest for several critical applications), the sharp stability profile that we obtain is new. Interestingly, we also prove that the (non-linear) profile for $r \leq \frac{N-1}{2}$ improves as $k$ becomes larger to such an extent that it becomes formally linear as $k$ goes to $\infty$. Finally, for any $k \geq 1$ and $0< \alpha \leq 1$, we show that our estimates are optimal within the class of $C^{k,\alpha}$ domains, by providing explicit examples.