Fourier series of the $\nabla div$ operator and Sobolev spaces II

Abstract

The author studies structure of space ${\mathbf{L}}_2(G)$ of vectors - functions, which are integrable with a square of the module on the bounded domain $G$of three-dimensional space with smooth boundary, and role of the gradient of divergence and curl operators in construction of bases in its orthogonal subspaces $\mathcal{A}$ and $\mathcal{B}$. The $\mathcal{A}$ and $\mathcal{B}$ are contain subspaces ${\mathcal{A}}_{\gamma }(G)\subset \mathcal{A}$ and ${\mathbf{V}}^0(G)\subset \mathcal{B}$. The gradient of divergence and a curl operators have continuations in these subspaces, their expansion ${\mathcal{N}}_d$ and $S$ are selfadjoint and convertible,and their inverse operators ${\mathcal{N}}_d^{-1}$ and $S^{-1}$ are compact. In each of these subspaces we build ortonormal basis. Uniting these bases, we receive complete ortonormal basis of whole space ${\mathbf{L}}_2(G)$, made from eigenfunctions of the gradient of divergence and curl operators . In a case, when the domain $G$ is a ball $B$, basic functions are defined by elementary functions. The spaces ${\mathcal{A}}_{\mathcal{K}}^s(B)$ are defined. Is proved, that condition $\mathbf{v}\in {\mathcal{A}}_{\mathcal{K}}^s(B)$ is necessary and sufficient for convergence of its Fourier series (on eigenfunctions of a gradient of divergence)in norm of Sobolev space ${\mathbf{H}}^s(B)$. Using Fourier series of functions $\mathbf{f}$ and $\mathbf{u}$, the author investigates solvability(in spaces ${\mathbf{H}}^s(G)$)boundary value problem: $\nabla \mathbf{div}\mathbf{u}+\lambda \mathbf{u}=\mathbf{f}$ in $G$, $\mathbf{n}\cdot \mathbf{u}{|}_{\Gamma }=g$ on boundary, under condition of $\lambda \ne 0$. In a ball $B$ a boundary value problem: $\nabla \mathbf{div}\mathbf{u}+\lambda \mathbf{u}=\mathbf{f}$ in $B$, $\mathbf{n}\cdot \mathbf{u}{|}_S=0$, is solved completely and for any $\lambda$.

in Russian