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

TL;DR

This paper develops Fourier series expansions for vector functions in Sobolev spaces using eigenfunctions of the gradient of divergence and curl operators, providing basis constructions, convergence criteria, and solutions to boundary value problems in three-dimensional domains.

Contribution

It introduces a complete orthonormal basis for vector Sobolev spaces based on eigenfunctions of key differential operators, and establishes convergence conditions and solvability results for related boundary value problems.

Findings

Constructed orthonormal bases from eigenfunctions of differential operators.

Proved convergence of Fourier series in Sobolev space norms.

Solved boundary value problems explicitly in the case of a ball domain.

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 $\mathcal{A}^{s}_{\mathcal{K}}(B)$ are defined. Is proved, that condition $ \mathbf{v}\in\mathcal{A}^{s}_{\mathcal {K}}(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 $\lambda\neq0$. 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 $\lambda$.