Isoperimetric Comparisons via Viscosity

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 supersolutions of some nonlinear differential equations. From these one can derive the isoperimetric inequalities of Lévy-Gromov and Bérard–Besson–Gallot, as well as an upper bound of Morgan–Johnson.