Concentration comparison for nonlinear diffusion on model manifolds and P\'olya-Szeg\H{o} inequality

TL;DR

This paper explores when the mass concentration comparison for nonlinear diffusion equations holds on spherically symmetric Riemannian manifolds, linking it to the validity of the Pólya-Szegő inequality and geometric conditions like scalar curvature.

Contribution

It establishes a precise equivalence between the concentration comparison and the Pólya-Szegő inequality on model manifolds, and identifies geometric conditions ensuring their validity.

Findings

Concentration comparison holds iff the Pólya-Szegő inequality holds on the manifold.

Supports for the Pólya-Szegő inequality include hyperbolic space and the sphere.

Scalar curvature conditions can cause the inequality to fail.

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 P\'olya-Szeg\H{o} 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 P\'olya-Szeg\H{o} 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 $ \phi: [0,+\infty) \to [0,+\infty) $, the filtration equation $ \partial_t u = \Delta \phi(u) $ satisfies the concentration comparison in $ \mathbb{M}^n \times (0,+\infty) $ if and only if $ \mathbb{M}^n $ supports the P\'olya-Szeg\H{o} 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 P\'olya-Szeg\H{o} inequality, and thus the concentration comparison, holds. This allows us to include important examples such as the hyperbolic space and the sphere.