Revisiting the Rellich inequality

TL;DR

This paper analyzes the Rellich inequality by separating radial and spherical derivatives, improving constants in the three-dimensional case, and comparing different Laplacian norms using spherical harmonics.

Contribution

It introduces a new perspective on the Rellich inequality by isolating derivative contributions and improves the best constant for the three-dimensional Laplace-Beltrami case.

Findings

Identified the best constant for the three-dimensional Laplace-Beltrami operator.

Compared norms of radial Laplacian and Laplace-Beltrami with the standard Laplacian.

Built on recent identities and spherical harmonics to refine inequalities.

Abstract

We revisit the Rellich inequality from the viewpoint of isolating the contributions from radial and spherical derivatives. This naturally leads to a comparison of the norms of the radial Laplacian and Laplace{Beltrami operators with the standard Laplacian. In the case of the Laplace{ Beltrami operator, the three-dimensional case is the most subtle and here we improve a result of Evans and Lewis by identifying the best constant. Our arguments build on certain identities recently established by Wadade and the second and third authors, along with use of spherical harmonics.