A Talenti-type comparison theorem for RCD(K,N) spaces and applications

We prove pointwise and Lp-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 RCD(K,N) metric measure space, with K> 0 and N∈ (1 , ∞)). The obtained Talenti-type comparison is sharp, rigid and stable with respect to L2/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 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.