Concentration comparison for nonlinear diffusion on model manifolds and Pólya-Szegő inequality

Abstract

We investigate the validity of the mass concentration comparison for a class of nonlinear diffusion equations posed on Riemannian manifolds ${\mathbb{M}}^n$ that are spherically symmetric, that is, model manifolds. The concentration comparison states that the solution of a certain diffusion equation that takes the radially decreasing (Schwarz) rearrangement $u_0^{\star }$ as its initial datum is more concentrated than the original solution starting from $u_0$. This is known to hold in ${\mathbb{R}}^n$ as a consequence of the celebrated Pólya-Szegő inequality, which asserts that the $L^2$ norm of the gradient of a function $f$ (belonging to an appropriate Sobolev space) is always larger than the $L^2$ norm of the gradient of its radially decreasing rearrangement $f^{\star }$. However, if ${\mathbb{M}}^n$ is a general model manifold, it is not for granted that the Pólya-Szegő inequality holds; in fact, we will provide a simple condition involving the scalar curvature of ${\mathbb{M}}^n$ under which such an inequality actually fails. The main result we prove states that, given any continuous, nondecreasing, and nontrivial function $\ddot{I}•:[0,+\infty )\to [0,+\infty )$, the filtration equation ${\partial }_{t}u=\hat{I}”\ddot{I}•(u)$ satisfies the concentration comparison in ${\mathbb{M}}^n\times (0,+\infty )$ if and only if ${\mathbb{M}}^n$ supports the Pólya-Szegő inequality. In particular, the validity of such a comparison for the heat equation is sufficient to guarantee that the same holds for all filtration equations. Moreover, we prove that if ${\mathbb{M}}^n$ supports a centered isoperimetric inequality then the Pólya-Szegő inequality, and thus the concentration comparison, holds. This allows us to include important examples such as the hyperbolic space and the sphere.