Operator gradient of divergencie in subspaces of ${\mathbf{L}}_2(G)$ space

Abstract

The author studies the structure of space ${\mathbf{L}}_2(G)$ of vector-valued functions that are square integrable in a bounded connected domain $G$ of the three-dimensional space with a smooth boundary and the role of gradient divergence operators and the rotor in the construction of bases in subspaces $\mathcal{A}$ and $\mathcal{B}$. The self-adjointness of the extension ${\mathcal{N}}_d$ of operator $\nabla \mathrm{div}$ to the subspace ${\mathcal{A}}_{\gamma }\subset \mathcal{A}$ and the basicity system of its own functions. Written explicit formulas for solving the spectral problem in a ball and the conditions for the decomposition vector-functions in a Fourier series in eigenfunctions gradient of divergence. The solvability of the boundary tasks: $\nabla \mathrm{div}\mathbf{u}+\lambda \mathbf{u}=\mathbf{f}$ in $G$, $(\mathbf{n}\cdot \mathbf{u}){|}_{\Gamma }=g$ in Sobolev spaces ${\mathbf{H}}^s(G)$ of order $s\ge 0$ and in subspaces. In passing, similar results for the operator of the rotor and its symmetric extension $S$ to $\mathcal{B}$.

in Russian