Weighted isoperimetric inequalities and some applications

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)].