Calculus of Variations and Geometric Measure Theory

D. Bartolucci - C. Gui - A. Jevnikar - A. Moradifam

A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

created by jevnikar on 27 Feb 2018
modified on 26 Sep 2018


Accepted Paper

Inserted: 27 feb 2018
Last Updated: 26 sep 2018

Journal: Math. Ann.
Year: 2018


We derive a singular version of the Sphere Covering Inequality which was recently introduced in 42, suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in 56. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.

Keywords: uniqueness results, Geometric PDEs, Mean field equation, Sphere covering inequality, Singular Liouville-type equations, Alexandrov-Bol inequality