Talenti's comparison theorem for Poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature

In this article, we prove Talenti’s comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $L^1$- and $L^{\mathrm{\infty }}$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse H"older inequality for eigenfunctions of Dirichlet Laplacian.