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

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-Pólya 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.

arXiv comment35 pages. Final version to appear in Calculus of Variations and Partial Differential Equations

References in corpus (7)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• Independence on $p$ of weak upper gradients on RCD spaces
• Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below
• Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds

Cited by in corpus (2)

• A reverse Hölder inequality for first eigenfunctions of the Dirichlet Laplacian on RCD(K,N) spaces
• Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability