# Comparison results for eigenvalues of curl curl operator and Stokes operator

Research paper by Zhibing Zhang

Indexed on: 20 Jul '18Published on: 19 Jul '18Published in: Zeitschrift für angewandte Mathematik und Physik

#### Abstract

This paper mainly establishes comparison results for eigenvalues of $${{\mathrm{curl}}}{{\mathrm{curl}}}$$ operator and Stokes operator. For three-dimensional simply connected bounded domains, the kth eigenvalue of $${{\mathrm{curl}}}{{\mathrm{curl}}}$$ operator under tangent boundary condition or normal boundary condition is strictly smaller than the kth eigenvalue of Stokes operator. For any dimension $$n\ge 2$$, 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 $${{\mathrm{curl}}}{{\mathrm{curl}}}$$ operator under tangent boundary condition or normal boundary condition is strictly larger than the second eigenvalue of Neumann Laplacian.