内插估计及其对一些定量对称结果的应用
Interpolating estimates with applications to some quantitative symmetry results
摘要：
我们证明了插值估计提供了一个边界的振荡功能方面的两个$L^p$规范的梯度。它们是基于锥上的函数的梯度的里斯位势的逐点界。这些估计适用于一般类别的领域，例如，Lipschitz域所有涉及的常数都可以显式计算。作为一个应用，我们展示了如何使用这些估计，以获得稳定的Alexandrov的肥皂泡定理和Serrin的超定边值问题。新的方法导致这些问题的几个新奇和好处。
英文摘要：
We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $L^p$ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.
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