A sharp weighted Wirtinger inequality

TL;DR

The paper investigates a sharp estimate for the best constant C in the weighted Wirtinger-type inequality ∫_0^{2π} γ^p w^2 ≤ C ∫_0^{2π} γ^q w'^2 for 2π-periodic functions w satisfying ∫_0^{2π} γ^p w = 0, with γ bounded above and below away from zero and p+q ≥ 0. They derive the smallest C for which the inequality holds under these hypotheses, i.e., a sharp constant. This result generalizes an inequality of Piccinini and Spagnolo to a broader weighted setting, by allowing γ^p and γ^q as the left and right weights. The findings provide optimal bounds for weighted periodic problems and may inform sharp weighted Sobolev-type estimates in applications.

Abstract

We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality \[ \int_0^{2\pi}\gamma^pw^2\le C\int_0^{2\pi}\gamma^qw'^2 \] where $\gamma$ is bounded above and below away from zero, $w$ is $2\pi$-periodic and such that $\int_0^{2\pi}\gamma^pw=0$, and $p+q\ge0$. Our result generalizes an inequality of Piccinini and Spagnolo.

Theorems & Definitions (7)

• Theorem 1
• Lemma 1: PS
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm:wirtinger']}