Sobolev spaces and operators vorticity and the gradient of the divergence

Abstract

In a bounded domain $G$ with smooth border studied boundary value and spectral problems for operators of the rotor (vortex) and the gradient of the divergence $+\lambda\,I$ in the Sobolev spaces. For $\lambda\neq 0$ these operators are reducible ( by B. Veinberg and V. Grushin method) to elliptical matrices and the boundary value problems satisfy the conditions of V. Solonnikov's ellipticity. Useful properties of solutions of these spectral problems follow from the theory and estimates. The $\nabla \text{div}$ and $ \text{rot}$ operators have self-adjoint extensions $\mathcal{N}_d$ and $\mathcal{S}$ in orthogonal subspaces $\mathcal{A}_{\gamma }$ and $\mathbf{V}^0$ which formed from potential and vortex fields in $\mathbf{L}_{2}(G)$. Their eigenvectors forme orthogonal basis in $\mathcal{A}_{\gamma }$ and $\mathbf{V}^0$ elements of which are presented by Fourier series and operators are transformations of series. We define analogues of Sobolev spaces $\mathbf{A}^{2k}_{\gamma }$ and $\mathbf{W}^m$ orders of $2k$ and $m$ in classes of potential and vortex fields and classes $ C (2k,m)$ of their direct sums. It is proved that if $\lambda\neq Sp(\mathrm{rot})$ the operator $ \text{rot}+\lambda\,I$ displays the class $C(2k,m+1)$ on the class $C(2k,m)$ one-to-one and continuously. And if $\lambda\neq Sp(\nabla \mathrm{div})$ operator $\nabla \text{div}+\lambda\,I$ maps class $C(2(k+1), m)$ on the class $C(2k,m)$, respectivly.