Optimal quantitative stability estimates for Alexandrov’s Soap Bubble Theorem via Gagliardo-Nirenberg-type interpolation inequalities
Year: 2026
Type: article
Abstract: The paper provides optimal quantitative stability estimates for the celebrated Alexandrov’s Soap Bubble Theorem within the class of $$C^{k,\alpha }$$ C k , α domains, for any $$k \ge 1$$ k ≥ 1 and $$0 < \alpha \le 1$$ 0 < α ≤ 1 , by leveraging Gagliardo-Nirenberg-type interpolation inequalities. Optimal estimates of uniform closeness to a ball are established for $$L^r$$ L r deviations of the mean curvature from being constant, for any $$r\ge 2$$ r ≥ 2 (more generally, for any $$r>1$$ r > 1 such that $$r\ge (2N-2)/(N+1)$$ r ≥ ( 2 N - 2 ) / ( N + 1 ) ). For $$r>\frac{N-1}{2}$$ r > 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 $$r\le \frac{N-1}{2}$$ r ≤ N - 1 2 , for which the profile is not linear, are new; even in the particular case $$r=2$$ 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 $$r \le \frac{N-1}{2}$$ r ≤ 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 $$k \ge 1$$
Authors: João Gonçalves da Silva, Giorgio Poggesi
Institutions: Adelaide University, The University of Adelaide
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4416358416
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