Sharp quantitative Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula
Sharp quantitative Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula
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尖锐定量 Faber-Krahn 不等式和 Alt-Caffarelli-Friedman 单调性公式
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发表时间：
2021
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通讯作者：
Robin Neumayer
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作者：
M. Allen;D. Kriventsov;Robin Neumayer
本文的目的有两个方面。首先，我们建立了单连通空间形式上Faber-Krahn不等式的新的精确的定量估计。我们证明了给定集合$Omega$的第一特征值与球的第一特征值之间的差距定量地控制了该集合到球{it}的L^1 $距离和相应特征函数之间的L^2 $距离：[Omega 1（Omega）-Omega 1（B）gtrsim|欧米茄德尔塔B| ^2 + int| u_{Omega} - u_B|其中，$B$表示与$Ω $最近的测地线球，其中，|B| =| Omega| $和$u_Omega$表示具有适当归一化的第一本征函数。在欧几里得空间，这扩展了Brasco-De Phillipis-Velichkov的结果;本征函数控制在很大程度上建立在我们的同伴论文中临界扰动Alt-Cafarelli型泛函极小化的新正则性结果上。在圆形球面和双曲空间，目前的结果是第一个尖锐的定量结果相对于任何距离，在这里的本地部分的分析是基于新的隐式谱分析技术。其次，我们应用这些尖锐的定量Faber-Krahn不等式，以建立一个定量形式的Alt-Caffarelli-Friedman（ACF）单调性公式。我们表明，从一个尺度到下一个ACF单调性公式中的能量下降控制如何接近一对容许函数是从一对互补的半平面解决方案。特别是，当所有尺度上的能量下降的平方根都很小时，我们的结果意味着这些函数的切线（唯一爆破）的存在。
The objective of this paper is two-fold. First, we establish new sharp quantitative estimates for Faber-Krahn inequalities on simply connected space forms. We prove that the gap between the first eigenvalue of a given set $Omega$ and that of the ball quantitatively controls both the $L^1$ distance of this set from a ball {it and} the $L^2$ distance between the corresponding eigenfunctions: [ lambda_1(Omega) - lambda_1(B) gtrsim |Omega Delta B|^2 + int |u_{Omega} - u_B|^2, ] where $B$ denotes the nearest geodesic ball to $Omega$ with $|B|=|Omega|$ and $u_Omega$ denotes the first eigenfunction with suitable normalization. On Euclidean space, this extends a result of Brasco-De Phillipis-Velichkov; the eigenfunction control largely builds upon new regularity results for minimizers of critically perturbed Alt-Cafarelli type functionals in our companion paper. On the round sphere and hyperbolic space, the present results are the first sharp quantitative results with respect to any distance; here the local portion of the analysis is based on new implicit spectral analysis techniques. Second, we apply these sharp quantitative Faber-Krahn inequalities in order to establish a quantitative form of the Alt-Caffarelli-Friedman (ACF) monotonicity formula. We show that the energy drop in the ACF monotonicity formula from one scale to the next controls how close a pair of admissible functions is from a pair of complementary half-plane solutions. In particular, when the square root of the energy drop summed over all scales is small, our result implies the existence of tangents (unique blowups) of these functions.