Fourier series of the ∇ div operator and Sobolev spaces II

2017/12/07 by Saks, R. S.
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1712.03804

Abstract

The author studies structure of space L₂(G) of vectors - functions, which are integrable with a square of the module on the bounded domain G of three-dimensional space with smooth boundary, and role of the gradient of divergence and curl operators in construction of bases in its orthogonal subspaces A and B. The A and B are contain subspaces A_γ(G)⊂A and V⁰(G)⊂B. The gradient of divergence and a curl operators have continuations in these subspaces, their expansion N_d and S are selfadjoint and convertible,and their inverse operators N_d^-1 and S^-1 are compact. In each of these subspaces we build ortonormal basis. Uniting these bases, we receive complete ortonormal basis of whole space L₂(G), made from eigenfunctions of the gradient of divergence and curl operators . In a case, when the domain G is a ball B, basic functions are defined by elementary functions. The spaces A^s_K(B) are defined. Is proved, that condition v\inA^s_\mathcal K(B) is necessary and sufficient for convergence of its Fourier series (on eigenfunctions of a gradient of divergence)in norm of Sobolev space H^s(B). Using Fourier series of functions f and u, the author investigates solvability(in spaces H^s(G))boundary value problem: \nabladivu+ λu=f in G, n\cdotu|_Γ=g on boundary, under condition of λ≠0. In a ball B a boundary value problem: \nabladivu+ λu=f in B, n\cdotu|_S=0, is solved completely and for any λ.

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