# Fourier series of the curl operator and Sobolev spaces

## TL;DR

This paper analyzes the spectral properties of curl and divergence operators in 3D domains, providing formulas for boundary value problems and conditions for Fourier series decompositions in Sobolev spaces.

## Contribution

It introduces exact formulas for solving boundary value problems and establishes conditions for Fourier series decompositions involving curl and divergence operators.

## Key findings

- Eigenfunctions form a basis in certain subspaces
- Exact formulas for boundary value problems in a ball
- Conditions for vector function decompositions

## Abstract

The properties of curl and gradient of divergence operators in the domain $G$ of three-dimensional space are described. The self-conjugacy of these operators in the subspaces $\mathbf{L}_{2}(G) $ and the basis property of the system of eigenfunctions are discussed. Exact formulas are founded for solving boundary value problems in a ball and the conditions for the decomposition of vector functions into Fourier series in eigenfunctions of the curl and the gradient of divergence operators.