Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A sharp weighted anisotropic Poincaré inequality for convex domains

Francesco Della Pietra, Nunzia Gavitone, Gianpaolo Piscitelli

Abstract

We prove an optimal lower bound for the best constant in a class of weighted anisotropic Poincaré inequalities

A sharp weighted anisotropic Poincaré inequality for convex domains

Abstract

We prove an optimal lower bound for the best constant in a class of weighted anisotropic Poincaré inequalities

Paper Structure

This paper contains 3 sections, 3 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Proof of the Payne-Weinberger inequality

Key Result

Theorem 1.1

Let $\mathcal{H}\in\mathscr H(\mathbb R^{n})$, $\mathcal{H}^{o}$ be its polar function. Let us consider a bounded convex domain $\Omega\subset\mathbb R^n$, $1<p<\infty$, and take a positive $\log$-concave function $\omega$ defined in $\Omega$. Then, given it holds that where $D_\mathcal{H}(\Omega)=\sup_{x,y\in\Omega}\mathcal{H}^{o}(y-x)$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Remark 3.3
  • Example 1