论文标题
肥皂气泡和凸锥：最佳定量刚性
Soap bubbles and convex cones: optimal quantitative rigidity
论文作者
论文摘要
我们考虑一类刚度导致凸锥$σ\ subseteq \ mathbb {r}^n $。这些包括相对于$σ$，Alexandrov的肥皂泡型结果相对于$σ$，与$σ$相对于$σ$的混合边界价值问题的过度确定性问题以及Heintze-Karcher相对于$σ$的不等式。每个刚性结果都是通过单个积分身份获得的，并在弱积分条件下保持真实。根据$ l^2 $ -PSEUDESTASE获得最佳定量稳定性估计。特别是，即使在经典情况下，Heintze-Karcher的不等式的最佳稳定性估计也是新的，即 还提供了Hausdorff距离的稳定性界限。 建立和利用了几个新的结果，包括针对矢量领域的新庞加莱型不平等，其正常组件在边界的一部分上消失了，并且具有明确的（可能是加权的）痕量理论 - 相对于锥体$σ$ - 用于谐波功能的圆锥$σ$ - 满足neumann neumann and of neumann neumann and of Borge and neumann and of Borge and of Borge neumann function in Borge and contion in neumann coption。 我们还引入了相对于锥体$σ\ subseteq \ mathbb {r}^n $的均匀内部和外部球体条件的新概念，这些概念允许（通过屏障参数）在混合边界值设定中获得（通过屏障参数）均匀的下限和上限。在特定情况下，$σ= \ mathbb {r}^n $，这些条件返回经典统一的内部和外部球体条件（以及迪里奇设置的相关经典梯度边界）。
We consider a class of rigidity results in a convex cone $Σ\subseteq \mathbb{R}^N$. These include overdetermined Serrin-type problems for a mixed boundary value problem relative to $Σ$, Alexandrov's soap bubble-type results relative to $Σ$, and a Heintze-Karcher's inequality relative to $Σ$. Each rigidity result is obtained by means of a single integral identity and holds true under weak integral conditions. Optimal quantitative stability estimates are obtained in terms of an $L^2$-pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher's inequality is new even in the classical case $Σ= \mathbb{R}^N$. 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 $Σ$ -- for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in $\partial Σ$. We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone $Σ\subseteq \mathbb{R}^N$, 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 $Σ= \mathbb{R}^N$, these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).
