A note on Poincar\'e-Sobolev type inequalities on compact manifolds

TL;DR

This paper establishes a new Poincaré-Sobolev inequality on compact manifolds, linking function deviations from a density-weighted average to the gradient's Lebesgue norm, with implications for elliptic systems.

Contribution

It introduces a novel inequality where the density influences the average but not the measure, providing a quantitative dependence of the Poincaré constant on the density's norm.

Findings

Proved a Poincaré-Sobolev inequality on compact manifolds.

Demonstrated the Poincaré constant depends on the density's Lebesgue norm.

Applicable to analysis of coupled elliptic systems.

Abstract

We prove a Poincar\'e-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical weighted Poincar\'e inequalities, the density does not enter the measure or the Sobolev norms, but only the reference average. We show that the associated Poincar\'e constant depends quantitatively on the Lebesgue norm of the density. This framework naturally arises in the analysis of coupled elliptic systems and seems not to have been addressed in the existing literature.