Sobolev spaces and boundary-value problems for the curl and gradient-of-divergence operators

We study boundary value and spectral problems in a bounded domain $G$ with smooth border for operators $\operatorname{rot} +\lambda I$ and $\nabla \operatorname{div} +\lambda I$ 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 and satisfy the conditions of V. Solonnikov's ellipticity. Useful properties of solutions of these spectral problems derive from the theory and estimates. The $\nabla \operatorname{div}$ and $ \operatorname{rot}$ operators have self-adjoint extensions $\mathcal{N}_d$ and $\mathcal{S}$ in orthogonal subspaces $\mathcal{A}_{\gamma }$ and $\mathbf{V}^0$ forming from potential and vortex fields in $\mathbf{L}_{2}(G)$. Their eigenvectors form orthogonal basis in $\mathcal{A}_{\gamma }$ and $\mathbf{V}^0$ elements 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 $\lambda\neq \operatorname{Sp}(\operatorname{rot})$, then the operator $ \operatorname{rot}+\lambda I$ displays the class $C(2k,m+1)$ on the class $C(2k,m)$ one-to-one and continuously. And if $\lambda\neq \operatorname{Sp}(\nabla \operatorname{div})$, then operator $\nabla \operatorname{div}+\lambda I$ maps the class $C(2(k+1), m)$ on the class $C(2k,m)$, respectively.

Sobolev spaces, gradient operator, divergence operator, curl operator, elliptic boundary value problems, spectral problems