Weighted isoperimetric inequalities and some applications

经典的等周不等式是欧几里德空间中一个集合的周长和体积之间的关系。在相同体积的比赛中，球的周长是独一无二的。这个几何不等式包含了许多解析不等式。特别地，它需要在施泰纳或施瓦茨对称下的Pólya-Szegö不等式。这可以用来得到Sobolev不等式的最佳常数。一个应用是Rayleigh-Faber-Krahn不等式，它说球在相同体积的域中具有最小的狄利克雷基态特征值。这是光谱几何和形状优化领域的基本结果。近年来，人们的注意力集中在测量周长和体积相对于密度时会发生什么。体积密度和周长密度可能相同，也可能不同。具有密度的等周问题的存在性和有界性结果包含在[Morgan and Pratelli(2013)]和[De Philippis， Franzina, Pratelli(2015)]中。在具有径向周长和体积权重的平面情况下，[McGillivray(2021)]中包含了第一个提到的结果的模拟。目的是在密度相对于度规从下面收敛（在第一个实例的径向情况下）的双加权情况下获得第二个提到的结果的模拟。log-凹密度猜想已经在[Chambers(2015)]中得到证明，并且（粗略地）表明，如果密度是log-凸的，欧几里得空间中的中心球是等周最小的。在平面情况下具有较弱假设的版本包含在[McGillivray(2018)]中。双曲平面上的对应项包含在[McGillivray(2019)]中。目的是探索是否在被击穿的球体上也有一些对应物。对数凸密度猜想是出于稳定性考虑而产生的。我们希望类似的方法可以对穿孔球产生相应的猜想。在双加权情况下的一个特殊情况是体积和周长的权重是径向幂。对于某些指数，中心球是唯一等周的。我们参考[Alvino， Brock, Chiacchio, Mercaldo and Posteraro(2017)]。后者在[McGillivray(2021)]中得到扩展，证明了其中和[Diaz， Harman, Howe, Thompson(2012)]中包含的一个猜想。在Alvino等人的论文中得到了一个相应的Pólya-Szegö不等式以及一个Caffarelli-Kohn-Nirenberg不等式（带权的Sobolev不等式的对应）。目的是在[McGillivray(2021)]所涵盖的情况下获得这些解析不等式的类似物。

The classical isoperimetric inequality is a relation between the perimeter and volume of a set in Euclidean space. Balls minimise perimeter uniquely amongst competitor sets with the same volume. This geometric inequality entails a number of analytic inequalities. In particular it entails the Pólya-Szegö inequality under Steiner or Schwarz symmetrisation. This can be used in turn to obtain the best constant in the Sobolev inequality. An application is the Rayleigh-Faber-Krahn inequality which says that the ball has smallest Dirichlet ground state eigenvalue amongst domains with the same volume. This is a basic result in the fields of Spectral Geometry and Shape Optimisation.In recent years attention has focused on what happens when both perimeter and volume are measured with respect to a density. The volume and perimeter densities may be the same or different.Existence and boundedness results for the isoperimetric problem with density are contained in [Morgan and Pratelli (2013)] and [De Philippis, Franzina, Pratelli (2015)]. An analogue of the first-mentioned result is contained in [McGillivray (2021)] in the planar case with radial perimeter and volume weights. An aim is to obtain an analogue of the second-mentioned result in the two-weighted situation when the density with respect to the metric converges from below (in the radial case in the first instance).The log-concave density conjecture has been proved in [Chambers (2015)] and states (roughly) that centred balls in Euclidean space are isoperimetric minimisers if the density is log-convex. A version with weaker hypotheses in the planar case is contained in [McGillivray (2018)]. A counterpart on the hyperbolic plane is contained in [McGillivray (2019)]. An aim is to explore whether there is also some counterpart on the punctured sphere. The log-convex density conjecture arose out of stability considerations. It is hoped that a similar approach might yield a corresponding conjecture for the punctured sphere. A particular case in the two-weighted situation is when the volume and perimeter weights are radial powers. For certain exponents centred balls are uniquely isoperimetric. We refer to [Alvino, Brock, Chiacchio, Mercaldo and Posteraro (2017)]. This last was extended in [McGillivray (2021)] proving a conjecture contained there and in [Diaz, Harman, Howe, Thompson (2012)]. A corresponding Pólya-Szegö inequality is obtained in the paper of Alvino et al as well as a Caffarelli-Kohn-Nirenberg inequality (a counterpart of the Sobolev inequality with weights). An aim is to obtain analogues of these analytic inequalities in the case covered in [McGillivray (2021)].