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

Abstract

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 ∞ -moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.

A reverse Hölder inequality for first eigenfunctions of the Dirichlet Laplacian on RCD(K,N) spaces

“…• smooth Riemannian manifolds with non-negative Ricci curvature and positive asymptotic volume ratio by Chen-Li [CL21].…”

Comparison results for Poisson equation with mixed boundary condition on manifolds

“…Talenti's comparison results play an important role in both partial differential equations and geometry problems, for instance there provide L ∞ estimate for solutions to PDEs and Faber-Krahn type inequality for the first Dirichlet eigenvalue, see [8,14,16]. Talenti's comparison results were generalized to nonlinear elliptic and parabolic equations with Dirichlet boundary condition (see for instance [4,5,6,20] and references therein), to compact Riemannian manifolds with positive Ricci curvature [11], and to noncompact manifold with nonnegative Ricci curvature and positive asymptotic volume ratio [8] as well. We also refer the reader to the excellent books [15,17] for related topics.…”

Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds

“…For Poisson equations with Dirichlet boundary conditions in Euclidean spaces, Talenti [25] gave pointwise comparisons of u ♯ and v. Talenti's comparison results were generalized to semilinear and nonlinear elliptic equations, for instance, in [26,3,15,24]. Recently, Talenti's comparison results were extended to solutions of Poisson equations on complete noncompact Riemannian manifolds with nonnegative Ricci curvature by the first two authors in [13]. We also refer the reader to excellent books [5,21,23] for related topics.…”

“…For the torsion problem with Dirichlet boundary condition on Riemannian manifold (M, g) satisfying Ric(g) ≥ (n − 1), the inequality (1.12) is due to [15] and [19]. In [13], the first two authors obtained sharp estimates for the L 1 -moment spectrum and L ∞ -moment spectrum with Dirichlet boundary conditions on bounded domains in complete Riemannian manifolds with nonnegative Ricci curvature.…”