A remark on optimal weighted Poincar\'e inequalities for convex domains

TL;DR

This paper establishes a precise upper bound for the best constant in weighted Poincaré inequalities on convex domains, depending solely on the diameter, through analysis of an optimal weighted Wirtinger inequality.

Contribution

It provides a sharp upper bound for weighted Poincaré inequalities on convex domains based only on diameter, advancing understanding of these inequalities.

Findings

Derived a sharp upper bound for the Poincaré constant

Connected the bound to the diameter of convex domains

Analyzed an optimal weighted Wirtinger inequality

Abstract

We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincar\'e inequalities. The key point is the study of an optimal weighted Wirtinger inequality.