2016/01/07 by S. V. Gryshchuk, Gryshchuk, S. V.
Computer Science · Mathematics · #30G35 #31A30 (Primary) #74B05 (Secondary) #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #FOS: Mathematics #Holomorphic and Operator Theory #Matrix Theory and Algorithms #math.AP #msc:30G35 #msc:31A30 #msc:74B05
paper · pdf · doi:10.48550/arxiv.1601.01626
arxiv created 2016/01/07 · openalex publication_date 2016/01/07 · arxiv updated 2016/01/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Consider the commutative algebra \mathbbB over the field of complex numbers with the bases \e1,e2\ such that %satisfying the conditions (e12+e22)2=0, e12+e22≠ 0. %\mathbbB is unique. Let D be a domain in xOy, Dζ:=\xe1+ye2:(x,y) ∈ D\⊂ \mathbbB. We say that \mathbbB-valued function Φ\colon Dζ \longrightarrow \mathbbB, Φ(ζ)=U1 e1+U2 ie1+ U3 e2+U4 ie2, ζ=xe1+ye2, Uk=Uk(x,y)\colon D\longrightarrow ℝ, k=1,4, is \em monogenic in Dζ iff Φ has the classic derivative in every point in Dζ. Every Uk, k=1,4, is a biharmonic function in D. A problem on finding an elastic equilibrium for isotropic body D by given boundary values on ∂ D of partial derivatives (∂ u)/(∂ v), (∂ v)/(∂ y) for displacements u, v is equivalent to BVP for monogenic functions, which is to find Φ by given boundary values of U1 and U4.