Solving of spectral problems for curl and Stokes operators

Abstract

In the work we explicitly solve the spectral problems for curl, gradient-divergence, and Stokes operators in a ball $B$ of radius $R$. The eigenfunctions $\mathbf{u}^{\pm}_{\kappa}$ of the curl associated with non-zero eigenvalues $\pm\lambda_{\kappa}$ are expressed by explicit formulas, as well as the vector-functions $\mathbf{q}_{\kappa}$ associated with the zero eigenvalue, \[rot \mathbf{u}^{\pm}_{\kappa}=\pm\lambda_{\kappa}  \mathbf{u}^{\pm}_{\kappa}, \quad \psi_n(\pm\lambda_{\kappa} R)=0, \quad \mathbf{n}\cdot\mathbf{u}^{\pm}_{\kappa}|_S=0;\quad rot \mathbf{q}_{\kappa}=0, \quad \mathbf{n}\cdot\mathbf{q}_{\kappa}|_S=0,\] where \[\psi_n(z)=(-z)^n(\frac{d}{zdz})^n\frac{\sin z}z, \quad \kappa=(n,m,k), n\geq 0,   m\in \mathbb{N},   |k|\leq n\] The same vector-functions are the eigenfunctions for the gradient-divergence operator with other eigenvalues, \[\nabla \mathrm{div} \mathbf{u}^{\pm}_{\kappa}=0; \quad \nabla \mathrm{div} \mathbf{q}_{\kappa}=\mu_{\kappa}\mathbf{q}_{\kappa}, \quad \mu_{\kappa}=(\alpha_{n,m}/R)^2,\quad \psi_n'(\alpha_{n,m})=0.\] The constructed system of vector eigenfunctions is complete and orthogonal in space ${\mathbf{{L}}_{2}}(B)$.The eigenfunctions $(\mathbf{v}_\kappa, \ p_\kappa)$ of the Stokes operator in the ball are represented as a sum of two eigenfunctions of the curl associated with opposite eigenvalues: ${\mathbf{v}_{\kappa }}= \mathbf{u}_{\kappa }^{+}+\mathbf{u}_{\kappa }^{-},$ $p_\kappa=\hbox{const}.$