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

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.