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

Daguang Chen, Haizhong Li|arXiv (Cornell University)|Apr 12, 2021

Geometric Analysis and Curvature Flows24 references4 citations

TL;DR

This paper establishes Talenti's comparison theorem for the Poisson equation on complete, noncompact Riemannian manifolds with nonnegative Ricci curvature and positive asymptotic volume ratio (AVR > 0). By adapting Schwarz symmetrization and leveraging the isoperimetric inequality under nonnegative Ricci curvature, the authors prove that the symmetric decreasing rearrangement of the solution on the manifold is pointwise dominated by the solution on a Euclidean ball of adjusted volume, with equality if and only if the manifold is Euclidean space and the domain is a ball.

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 $L^\infty$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.

Motivation & Objective

• To extend Talenti’s comparison theorem for the Poisson equation from Euclidean space to complete, noncompact Riemannian manifolds with nonnegative Ricci curvature.
• To establish a comparison principle using symmetrization that accounts for the manifold’s asymptotic volume ratio (AVR).
• To derive geometric and spectral inequalities—such as Faber-Krahn, Saint-Venant, and reverse Hölder—based on the generalized comparison theorem.
• To characterize equality cases in these inequalities, linking them to rigidity theorems (i.e., Euclidean space and spherical symmetry).

Proposed method

• Adapts Talenti’s original comparison principle to Riemannian manifolds using the asymptotic volume ratio (AVR) to scale the Euclidean comparison domain.
• Employs Schwarz symmetrization of the source function f and the solution u, mapping them to radially symmetric functions on a Euclidean ball of volume scaled by AVR.
• Applies the isoperimetric inequality on manifolds with nonnegative Ricci curvature (from [11]) to control the distribution of measure and energy.
• Uses the first eigenvalue comparison via symmetrization to derive the Faber-Krahn inequality, linking the first Dirichlet eigenvalue to the Euclidean model.
• Applies the method of moving planes and spectral comparison to prove reverse Hölder inequalities for eigenfunctions.
• Employs energy estimates and rearrangement inequalities to compare L^p norms of solutions and their symmetrized counterparts.

Experimental results

Research questions

• RQ1 Can Talenti’s comparison theorem for the Poisson equation be extended to complete, noncompact Riemannian manifolds with nonnegative Ricci curvature?
• RQ2 How does the asymptotic volume ratio (AVR) affect the symmetrization and comparison of solutions on such manifolds?
• RQ3 What spectral and geometric inequalities (e.g., Faber-Krahn, Saint-Venant) can be derived from the generalized comparison principle?
• RQ4 Under what conditions does equality hold in the comparison, and what does it imply about the manifold’s geometry?
• RQ5 Can reverse Hölder inequalities for eigenfunctions of the Dirichlet Laplacian be established via symmetrization on such manifolds?

Key findings

• The symmetric decreasing rearrangement of the solution u on a domain Ω in a manifold with nonnegative Ricci curvature and AVR > 0 is pointwise dominated by the solution v on a Euclidean ball Ω^♯ of volume scaled by AVR.
• Equality in the comparison u^♯(x) ≤ v(x) holds if and only if the manifold is isometric to Euclidean space and Ω is isoperimetric to Ω^♯.
• The Faber-Krahn inequality λ₁(Ω) ≥ λ₁(Ω^♯) holds, with equality if and only if (M,g) is Euclidean and Ω is a ball.
• The Saint-Venant theorem for torsional rigidity is generalized: the torsional rigidity of Ω is bounded above by that of Ω^♯, with equality under the same rigidity condition.
• A reverse Hölder inequality for eigenfunctions of the Dirichlet Laplacian is established: ||u||_{L^q(Ω)} ≤ AVR^{1/q} ||v||_{L^q(B_λ)} for q ≥ 1.
• The second eigenvalue λ₂(Ω) satisfies λ₂(Ω) > 2^{2/n} j_{n/2−1,1}^2 (ω_n AVR / |Ω|)^{2/n}, generalizing the Krahn–Hartman–Szegö inequality to non-Euclidean settings.