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

R. S. Saks

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 27 (2023) no. 1, pp. 23-49

Résumé

We will consider the scale of the Sobolev spaces $\mathbf{H}^{m}(G)$ vector fields in a bounded domain $G$ of $\mathbb{R}^3$ with a smooth boundary of $\Gamma$. The gradient-of-divergence and the rotor-of-rotor operators ($\nabla \,\text{div}$ and $ \text{rot}^2$) and their powers are analogous to the scalar operator $\Delta^m$ in $\mathbb{R}^3$. They generate spaces $ \mathbf{A}^{2k}(G)$ and $\mathbf{W}^m(G)$ potential and vortex fields; where the numbers $k$, $m>0$ are integers. It is proven that $ \mathbf{A}^{2k}(G)$ and $\mathbf{W}^m(G)$ are projections of Sobolev spaces $ \mathbf{H}^{2k}(G) $ and $ \mathbf{H}^{m}(G)$ in subspaces $\mathcal{A}$ and $\mathcal{B}$ in $\mathbf{L}_{2}(G)$. Their direct sums $ \mathbf{A}^{2k}(G) \oplus \mathbf{W}^m(G)$ form a network of spaces. Its elements are classes $ \mathbf{C}(2k, m)\equiv \mathbf{A}^{2k}\oplus \mathbf{W}^m$. We consider at the properties of the spaces $\mathbf{A}^{-m}$ and $\mathbf{W}^{-m}$ and proved their compliance with the spaces $\mathbf{A}^{m}$ and $\mathbf{W}^{m}$. We also consider at the direct sums of $ \mathbf{A}^{k}(G)\oplus \mathbf{W}^m(G)$ for any integer numbers $k$ and $m>0$. This completes the construction of the $\{\mathbf{C}(k, m)\}_{k,m}$ network. In addition, an orthonormal basis has been constructed in the space $\mathbf{L}_{2}(G)$. It consists of the orthogonal subspace $\mathcal{A}$ and $\mathcal{B}$ bases. Its elements are eigenfields of the operators $\nabla\,\text{div}$ and $\text{rot}$. The proof of their smoothness is an important stage in the theory developed. The model boundary value problems for the operators $\text{rot}+\lambda I$, $\nabla\,\text{div}+\lambda I $, their sum, and also for the Stokes operator have been investigated in the network $\{\mathbf{C}(k, m)\}_{k,m}$. Solvability conditions are obtained for the model problems considered.

Détail

Mots-clés : Sobolev spaces
Keywords: gradient operator, divergence operator, curl operator, elliptic boundary value problems, spectral problems.