2022/04/14 by R. S. Saks, Saks, Romen Semenovich
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2204.14225
openalex publication_date 2022/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The properties of the curl and the gradient of divergence operators ( rot and \nabladiv ) are studied in the space \mathbf L2 (G) in a bounded domain G ⊂ \textrm R3 with a smooth boundary Γ and in the classes C(2k, m)(G)≡ A2k(G) ⊕ Wm(G). The space \mathbf L2 (G) is decomposed into orthogonal subspaces A and \mathcal B : L2(G)=A⊕ B. In turn, A= AH⊕ A0 and B=BH ⊕ V0, where AH and BH are null spaces of operators ∇ div and rot in A and B; the dimensions of AH and BH are finite and determined by the topology of the boundary; AH=∅ and BH= ∅ if the domain Ω is a ball. The orthonormal basis are constructed in the class A0 (resp., In V0 ) by eigenfields qj(x) of ∇ div operator (resp., q± j(x) of rot operator) with nonzero eigenvalues μj (resp., ± λj ). The operators \nabladiv and rot cancel each other out and project L2(G) onto \mathcal A and \mathcal B , and \mathrm rot \mathbf u = 0 for \mathbf u ∈ \mathcal A , and ∇ \mathrm div \mathbf v = 0 for \mathbf v ∈ \mathcal B \citehw. Laplace matrix operator expressed through them: \mathrmΔ \mathbf v ≡ ∇ div \mathbf v -(rot)2 \mathbf v.