Differentiable functions on modules and the equation grad(w)=Mgrad(v)

Journal article

Let A be a finite-dimensional, commutative algebra over R or C. The notion of A-differentiable functions on A is extended to develop a theory of A-differentiable functions on finitely generated A-modules. Let U be an open, bounded and convex subset of such a module. An explicit formula is given for A-differentiable functions on U of prescribed class of differentiability in terms of real or complex differentiable functions, in the case when A is singly generated and the module is arbitrary and in the case when A is arbitrary and the module is free. Certain components of A-differentiable function are proved to have higher differentiability than the function itself.
Let M be a constant, square matrix. By using the formula mentioned above, a complete description of solutions of the equation grad(w) = Mgrad(v) is given.
A boundary value problem for generalized Laplace equations M∇2 v = ∇2 vMT is formulated and it is shown that for given boundary data there exists a unique solution, for which a formula is provided.

Authors: Ciosmak, KJ

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Mathematical Society
• Journal: St. Petersburg Mathematical Journal
• Volume: 34
• Issue: 2
• Pages: 271-303
• Publication date: 2023-03-22
• Acceptance date: 2021-12-20
• DOI: 10.1090/spmj/1754
• EISSN: 1547-7371
• ISSN: 1061-0022
• Language: English
• Keywords: generalised Laplace equations; generalised analytic functions; FFR; Banach algebra of A-differentiable functions; differentiable functions on algebraz