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

Abstract We prove pointwise and $$L^{p}$$ 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 an $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) metric measure space, $$K>0$$ > 0 $$N\in (1,\infty )$$ ? 1 ? ). The obtained Talenti-type is sharp, rigid stable respect $$L^{2}$$ 2 /measured-Gromov–Hausdorff topology; moreover, several aspects seem new even smooth Riemannian manifolds. As applications such comparison, we series improved Sobolev-type inequalities, $${{\,\mathrm{RCD}\,}}$$ version the St. Venant-Pólya torsional rigidity theorem (with associated stability statements). Finally, give probabilistic interpretation (in setting manifolds) aforementioned results, in terms exit time from subset Brownian motion.