Almost rigidity of the Talenti-type comparison theorem on $\mathrm{RCD}(0,N)$ space

TL;DR

This paper extends the Talenti comparison theorem to $p$-Laplacian problems on $ ext{RCD}(0,N)$ spaces and establishes an almost rigidity result, advancing understanding of geometric analysis in non-smooth spaces.

Contribution

It proves a Talenti-type comparison theorem for the $p$-Laplacian on $ ext{RCD}(0,N)$ spaces and introduces an almost rigidity result using compactness methods.

Findings

Established a Talenti-type comparison theorem on $ ext{RCD}(0,N)$ spaces.

Proved an almost rigidity result for the comparison theorem.

Utilized compactness techniques on varying spaces.

Abstract

In this paper, we prove a Talenti-type comparison theorem for the $p$-Laplacian with Dirichlet boundary conditions on open subsets of a $\mathrm{RCD}(0,N)$ space with $N\in (1,\infty)$. We also obtain an almost rigidity result of the Talenti-type comparison theorem, whose proof relies on a compactness on varying spaces.