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In the work we explicitly solve the spectral problems for curl, gradientdivergence, and Stokes operators in a ball B of radius R. The eigenfunctions uκ of the curl associated with non-zero eigenvalues ±λκ are expressed by explicit formulas, as well as the vector-functions qκ associated with the zero eigenvalue, rotuκ = ±λκ uκ , ψn(±λκR) = 0, n · uκ |S = 0; rotqκ = 0, n · qκ|S = 0, where ψn(z) = (−z) ( d zdz )n sin z z , κ = (n,m, k), n ≥ 0, m ∈ N, |k| 6 n The same vector-functions are the eigenfunctions for the gradient-divergence operator with other eigenvalues, ∇ divuκ = 0; ∇ divqκ = μκqκ, μκ = (αn,m/R), ψ′ n(αn,m) = 0. The constructed system of vector eigenfunctions is complete and orthogonal in space L2(B). The eigenfunctions (vκ, pκ) of the Stokes operator in the ball are represented as a sum of two eigenfunctions of the curl associated with opposite eigenvalues: vκ = u + κ + u − κ , pκ = const.

Solving of spectral problems for curl and Stokes operators