Comparison results for eigenvalues of curlcurl operator and Stokes operator

TL;DR

This paper establishes comparison results for eigenvalues of the curl-curl and Stokes operators, revealing inequalities between their eigenvalues in various domain settings and boundary conditions.

Contribution

It provides new eigenvalue comparison theorems for curl-curl and Stokes operators in different domain geometries and boundary conditions.

Findings

Eigenvalues of curl-curl are smaller than those of Stokes in 3D bounded domains.

First eigenvalue of Stokes exceeds that of Dirichlet Laplacian in any dimension.

In convex 3D domains, curl-curl eigenvalues are larger than Neumann Laplacian's second eigenvalue.

Abstract

This paper mainly establishes comparison results for eigenvalues of $\curl\curl$ operator and Stokes operator. For three-dimensional simply connected bounded domains, the $k$-th eigenvalue of $\curl\curl$ operator under tangent boundary condition or normal boundary condition is strictly smaller than the $k$-th eigenvalue of Stokes operator. For any dimension $n\geq2$, the first eigenvalue of Stokes operator is strictly larger than the first eigenvalue of Dirichlet Laplacian. For three-dimensional strictly convex domains, the first eigenvalue of $\curl\curl$ operator under tangent boundary condition or normal boundary condition is strictly larger than the second eigenvalue of Neumann Laplacian.