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.

我们证明了在具有正 Ricci 曲率的（可能是非光滑的）空间的开放子集上定义的椭圆狄利克雷问题的解决方案的逐点和\(L^{p}\)梯度比较结果（更准确地说是\({{\ ,\mathrm{RCD}\,}}(K,N)\)度量空间，用\(K>0\)和\(N\in (1,\infty )\)）。就\(L^{2}\) /measured-Gromov–Hausdorff 拓扑而言，获得的Talenti 型比较是尖锐、刚性和稳定的；此外，即使对于平滑的黎曼流形来说，有几个方面似乎是新的。作为这种Talenti型比较的应用，我们证明了一系列改进的Sobolev型不等式，以及\({{\,\mathrm{RCD}\,}}\)St. Venant-Pólya 扭转刚度比较定理的版本（以及相关的刚度和稳定性陈述）。最后，我们根据布朗运动的开放子集的退出时间，给出上述比较结果的概率解释（在平滑黎曼流形的设置中）。