Isoperimetric comparisons via viscosity

Abstract

Viscosity solutions are suitable notions in the study of nonlinear PDEs justified by estimates established via the maximum principle or the comparison principle. Here we prove that the isoperimetric profile functions of Riemannian manifolds with Ricci lower bound are viscosity super-solutions of some nonlinear differential equations. From these one can derive the isoperimetric inequalities of L\'evy-Gromov and B\'erard-Besson-Gallot, as well as some new comparison results.