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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.