Fourier series of the curl operator and Sobolev spaces

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.

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