Sobolev spaces and operators vorticity and the gradient of the divergence

TL;DR

This paper investigates boundary value and spectral problems for the rotor and gradient of divergence operators in Sobolev spaces within bounded domains, establishing their spectral properties, self-adjoint extensions, and Sobolev space analogues.

Contribution

It introduces new Sobolev space frameworks and analyzes spectral properties of rotor and divergence operators, including their self-adjoint extensions and spectral mappings.

Findings

Operators are reducible to elliptical matrices for nonzero λ.

Eigenvectors form orthogonal bases in potential and vortex subspaces.

Operators map specific Sobolev classes continuously under spectral conditions.

Abstract

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