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

TL;DR

This work addresses quantitative stability for Alexandrov's Soap Bubble Theorem by linking $L^r$-type mean curvature deviations to uniform spherical closeness within $C^{k,\alpha}$ domains using Gagliardo-Nirenberg-type interpolation. The authors derive an explicit stability exponent $\tau_{k,\alpha,N,r}=\frac{k+\alpha}{k+\alpha+\frac{N-1-2r}{r}}$, obtaining linear stability for $r>\frac{N-1}{2}$, a log-improved bound at the critical $r=\frac{N-1}{2}$, and nonlinear profiles for $r<\frac{N-1}{2}$; all exponents are shown to be sharp and the regime becomes formally linear as $k+\alpha\to\infty$. A key methodological contribution is reducing to nearly spherical sets when curvature deviation is small, enabling a dimension-reduction approach and the use of GN interpolation to derive sharp rates. The results generalize and sharpen existing stability bounds, providing optimal rates that depend on the regularity parameters $k$ and $\alpha$ and proving optimality through explicit constructions. Overall, the paper advances the understanding of stability in geometric PDEs and offers precise, optimal quantitative tools for proximity to a ball in Alexandrov-type problems.

Abstract

The paper provides optimal quantitative stability estimates for the celebrated Alexandrov's Soap Bubble Theorem within the class of $C^{k,α}$ domains, for any $k \ge 1$ and $0 < α\leq 1$, 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 $0< α\leq 1$, we show that our estimates are optimal within the class of $C^{k,α}$ domains, by providing explicit examples.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3: The case $r=\frac{N-1}{2}$
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Definition 2.1
• Lemma 2.2
• Theorem 2.3: Theorem 1, BrezisMironescu
• Corollary 2.4