Weighted isoperimetric inequalities and some applications

经典的等周不等式是欧氏空间中一个集合的周长和体积之间的关系。球最大限度地减少周长唯一的竞争对手之间的设置具有相同的体积。这个几何不等式包含了许多解析不等式。特别地，它需要在Steiner或施瓦茨对称化下的Pólya-Szegö不等式。这可以反过来用于获得Sobolev不等式中的最佳常数。一个应用是Rayleigh-Faber-Krahn不等式，它说球在具有相同体积的域中具有最小的Dirichlet基态本征值。这是谱几何和形状优化领域的一个基本结果。近年来，人们的注意力集中在当周长和体积都相对于密度测量时会发生什么。体积密度和周长密度可以相同或不同。密度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）]中包含了与第一个结果类似的结果。一个目的是在两个加权的情况下，当密度相对于度量从下收敛时（在第一种情况下是径向的情况下），得到第二个结果的类似结果。对数凹密度猜想已经在[Chambers（2015）]中得到证明，并且（粗略地）指出，如果密度是对数凸的，则欧几里得空间中的中心球是等周极小化器。平面情况下假设较弱的版本包含在[McGillivray（2018）]中。双曲平面上的对应物包含在[McGillivray（2019）]中。一个目的是探索是否也有一些对应的穿刺球。对数凸密度猜想是出于对稳定性的考虑。人们希望类似的方法可能会产生一个相应的猜想穿孔球。在两个加权的情况下的一个特殊情况是当体积和周长权重是径向功率时。对于一定的指数，中心球是唯一等周的。我们参考[Alvino，Brock，Chiacchio，Mercaldo和Posteraro（2017）]。这最后是延长在[麦吉利夫雷（2021）]证明一个猜想包含在那里和[迪亚兹，哈曼，豪，汤普森（2012）]。在Alvino等人的文章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）]中所涵盖的情况下获得这些分析不等式的类似物。

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)].