2016/07/19 by Krzysztof J. Ciosmak, Krzysztof Ciosmak, Ciosmak, Krzysztof
Mathematics · #46J15 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Analysis of PDEs (math.AP) #Boundary (topology) #Bounded function #Commutative property #Complex Variables (math.CV) #Convex function #Differentiable function #FOS: Mathematics #Function (biology) #Functional Analysis (math.FA) #Geometry #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Nabla symbol #Physics #Primary 30G35 #Pure mathematics #Regular polygon #Secondary 35N05 #math.AP #math.CV #math.FA #msc:30G35 #msc:35N05 #msc:46J15
paper · pdf · doi:10.48550/arxiv.1607.05624
Accepted in Algebra and Analysis, 2022 (St. Petersburg Mathematical Journal), 39 pages. Comments welcome
openalex publication_date 2016/07/19 · arxiv created 2022/02/07 · arxiv updated 2022/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a finite-dimensional, commutative algebra over ℝ or ℂ. The notion of A-differentiable functions on A is extended to the notion of A-differentiable functions on a finitely generated A-module B. Let U be an open, bounded and convex subset of B. When A is singly generated and B is arbitrary or A is arbitrary and B is a free module, an explicit formula for an A-differentiable functions on U, of a prescribed class of differentiability, is given in terms of real or complex differentiable functions. It appears, even in case of real algebras, that certain components of A-differentiable function are of higher differentiability than the function itself. Let M be a constant, square matrix. Using the aforementioned formula we find the complete solution of the equation grad(w)=Mgrad(v). The boundary value problem for generalized Laplace equations M∇2 v=∇2v M\intercal is formulated and it is proved that for the given boundary data there exists an unique solution, for which a formula is provided.