A Talenti-type comparison theorem for $\mathrm{RCD}(K,N)$ spaces and applications

TL;DR

This paper establishes sharp Talenti-type comparison theorems for solutions to elliptic problems on $ ext{RCD}(K,N)$ spaces, leading to improved inequalities and probabilistic interpretations, extending classical results to non-smooth spaces.

Contribution

It introduces a novel Talenti-type comparison theorem for $ ext{RCD}(K,N)$ spaces, with applications to inequalities and stochastic processes, even in non-smooth settings.

Findings

Sharp, rigid, and stable comparison results for elliptic solutions.

New Sobolev-type inequalities derived from the comparison.

Probabilistic interpretation via Brownian motion exit times.

Abstract

We prove pointwise and $L^{p}$-gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more precisely of an $\mathrm{RCD}(K,N)$ metric measure space, with $K>0$ and $N\in (1,\infty)$). The obtained Talenti-type comparison is sharp, rigid and stable with respect to $L^{2}$/measured-Gromov-Hausdorff topology; moreover, several aspects seem new even for smooth Riemannian manifolds. As applications of such Talenti-type comparison, we prove a series of improved Sobolev-type inequalities, and an $\mathrm{RCD}$ version of the St.~Venant-P\'olya torsional rigidity comparison theorem (with associated rigidity and stability statements). Finally, we give a probabilistic interpretation (in the setting of smooth Riemannian manifolds) of the aforementioned comparison results, in terms of exit time from an open subset for the Brownian motion.