Higher-order Sobolev and Rellich inequalities via iterated Talenti's principle
Csaba Farkas, Sándor Kajántó
TL;DR
The paper develops a systematic framework to extend Euclidean higher-order Sobolev and Rellich inequalities to complete non-compact Riemannian manifolds that satisfy a sharp isoperimetric inequality. Central to the approach is an iterated Talenti principle, implemented via Schwarz symmetrization and comparison of solutions to appropriately symmetrized PDEs, enabling transfer of Euclidean constants $S_{m,p}$ and $R_{m,p}$ to the geometric setting with the factor $(c_g)^{mp}$. The main results include: (i) higher-order Sobolev inequalities with constants $(c_g)^{mp} S_{m,p}$ and the corresponding sharpness in certain geometric regimes, (ii) higher-order Rellich inequalities with constants $(c_g)^{mp} R_{m,p}$ and improved variants using a nonincreasing weight $h$, all under the hypothesis $n>mp$. The work covers settings such as Brendle-type Euclidean volume growth, Cartan–Hadamard-type nonpositive curvature, and related rigidity/sharpness phenomena, contributing to a deeper understanding of geometric PDEs on manifolds and their optimal constants.
Abstract
In this paper we establish higher-order Sobolev and Rellich-type inequalities on non-compact Riemannian manifolds supporting an isoperimetric inequality. We highlight two notable settings: manifolds with non-negative Ricci curvature and having Euclidean volume growth (supporting Brendle's isoperimetric inequality) and manifolds with non-positive sectional curvature (satisfying the Cartan-Hadamard conjecture or supporting Croke's isoperimetric inequality). Our proofs rely on various symmetrization techniques, the key ingredient is an iterated Talenti's comparison principle. The non-iterated version is analogous to the main result of Chen and Li [J. Geom. Anal., 2023].