Berezin--Li--Yau inequalities on domains on the sphere

TL;DR

This paper establishes Berezin--Li--Yau inequalities for eigenvalues of Laplacians on domains on the sphere, with detailed analysis for the 2-sphere including vector Laplacians and the Stokes operator.

Contribution

It extends Berezin--Li--Yau inequalities to spherical domains and provides detailed results for the 2-sphere, including vector and Stokes operators.

Findings

Proved Berezin--Li--Yau inequalities on spherical domains

Derived specific results for the 2-sphere case

Included analysis of vector Laplacian and Stokes operator

Abstract

We prove Berezin--Li--Yau inequalities for the Dirichlet and Neumann eigenvalues on domains on the sphere $\mathbb{S}^{d-1}$. The case of $\mathbb{S}^{2}$ is treated in greater detail, including the vector Dirichlet Laplacian and the Stokes operator.