Soap bubbles and convex cones: optimal quantitative rigidity

IF 1.2 2区 数学 Q1 MATHEMATICS
Giorgio Poggesi
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引用次数: 0

Abstract

We consider a class of recent rigidity results in a convex cone Σ R N \Sigma \subseteq \mathbb {R}^N . These include overdetermined Serrin-type problems for a mixed boundary value problem relative to Σ \Sigma , Alexandrov’s soap bubble-type results relative to Σ \Sigma , and Heintze-Karcher’s inequality relative to Σ \Sigma . Each rigidity result is obtained here by means of a single integral identity and holds true under weak integral overdeterminations in possibly non-smooth cones. Optimal quantitative stability estimates are obtained in terms of an L 2 L^2 -pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher’s inequality is new even in the classical case Σ = R N \Sigma = \mathbb {R}^N .

Stability bounds in terms of the Hausdorff distance are also provided.

Several new results are established and exploited, including a new Poincaré-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory – relative to the cone Σ \Sigma – for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in Σ \partial \Sigma .

We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone Σ R N \Sigma \subseteq \mathbb {R}^N , which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case Σ = R N \Sigma =\mathbb {R}^N , these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).

肥皂泡和凸锥体:最佳定量刚性
我们考虑了凸锥 Σ \Sigma \subseteq \mathbb {R}^N 中的一类最新刚性结果。这些结果包括相对于 Σ \Sigma 的混合边界值问题的超定 Serrin 型问题,相对于 Σ \Sigma 的 Alexandrov 肥皂泡型结果,以及相对于 Σ \Sigma 的 Heintze-Karcher 不等式。这里的每一个刚性结果都是通过单一积分同一性得到的,并且在可能是非光滑圆锥的弱积分超定条件下都成立。最优定量稳定性估计是通过 L 2 L^2 伪距得到的。特别是,即使在经典情形 Σ = R N \Sigma = \mathbb {R}^N 中,海因策-卡尔切不等式的最优稳定性估计也是新的。还提供了豪斯多夫距离的稳定性边界。我们建立并利用了几个新结果,包括针对其法向分量在部分边界上消失的向量场的新的波恩卡莱式不等式,以及相对于锥Σ \Sigma 的明确(可能加权)迹理论,用于满足∂ Σ \partial \Sigma 所含部分边界上的同质诺依曼条件的谐函数。我们还引入了相对于圆锥 Σ ⊆ R N \Sigma \subseteq \mathbb {R}^N 的均匀内部和外部球面条件的新概念,从而可以(通过障碍论证)得到混合边界值设定中梯度的均匀下界和上界。在 Σ = R N \Sigma = \mathbb {R}^N 的特殊情况下,这些条件返回经典的均匀内部和外部球面条件(以及相关的经典梯度边界)。
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来源期刊
CiteScore
2.30
自引率
7.70%
发文量
171
审稿时长
3-6 weeks
期刊介绍: All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles in all areas of pure and applied mathematics. To be published in the Transactions, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Papers of less than 15 printed pages that meet the above criteria should be submitted to the Proceedings of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.
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