论文标题
$ \ mathrm {rcd}（k，n）$ space and应用程序的Talenti-Type比较定理
A Talenti-type comparison theorem for $\mathrm{RCD}(K,N)$ spaces and applications
论文作者
论文摘要
我们证明了点重点，$ l^{p} $ - 梯度比较结果的椭圆形dirichlet问题的解决方案在具有正ricci曲率的（可能是非平滑的）空间的开放子集上定义了（$ \ mathrm {rcd}（rcd}（k，k，n）$ nmetric Memalric Mature Space，$ k> $ n $ n $ $ n \ f。就$ l^{2} $/测量的Gromov-Hausdorff拓扑而言，获得的Talenti-Type比较是锋利，僵化和稳定的；此外，即使对于平滑的里曼尼亚人歧管，几个方面似乎都是新的。作为这种Talenti-Type比较的应用，我们证明了一系列改进的Sobolev型不平等，以及St.最后，就布朗运动的开放子集的退出时间而言，我们给出了上述比较结果的概率解释（在平滑的riemannian歧管的情况下）。
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.
