Operator gradient of divergencie in subspaces of L₂(G) space

2017/10/17 by Saks, R. S.
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1710.06428

Abstract

The author studies the structure of space \mathbf L _ 2 (G) of vector-valued functions that are square integrable in a bounded connected domain G of the three-dimensional space with a smooth boundary and the role of gradient divergence operators and the rotor in the construction of bases in subspaces \mathcal A and \mathcal B . The self-adjointness of the extension \mathcal N _d of operator ∇ \mathrm div to the subspace \mathcal A _ γ ⊂ \mathcal A and the basicity system of its own functions. Written explicit formulas for solving the spectral problem in a ball and the conditions for the decomposition vector-functions in a Fourier series in eigenfunctions gradient of divergence. The solvability of the boundary tasks: ∇ \mathrm div \mathbf u + λ \mathbf u = \mathbf f in G , (\mathbf n ⋅ \mathbf u) | _ Γ = g in Sobolev spaces \mathbf H ^ s (G) of order s ≥ 0 and in subspaces. In passing, similar results for the operator of the rotor and its symmetric extension S to \mathcal B .

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