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A Talenti-type comparison theorem for \(\mathrm{RCD}(K,N)\) spaces and applications. (English) Zbl 1480.53060

Summary: 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.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35R01 PDEs on manifolds
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