Best constants in Poincar\'e inequalities for convex domains

TL;DR

This paper establishes a sharp upper bound for the best constants in Poincaré inequalities on convex domains, using a refined rearrangement inequality to relate eigenvalues to the domain's diameter.

Contribution

It introduces a new sharp inequality for p-Laplacian Neumann eigenvalues on convex domains, refining classical rearrangement techniques for optimal bounds.

Findings

Proves a Payne-Weinberger type inequality for p-Laplacian eigenvalues.

Provides the sharp upper bound for Poincaré constants based on domain diameter.

Utilizes a refined Pólya-Szegő inequality for symmetric decreasing rearrangements.

Abstract

We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincar\'e inequality. The key point is the implementation of a refinement of the classical P\'olya-Szeg\"o inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.