Generalizations of the Dirichlet problem for bianalytic functions

TL;DR

This paper extends the Dirichlet problem solutions to bianalytic and bicomplex differential equations, demonstrating well-posedness under specific boundary conditions on non-circular conics.

Contribution

It generalizes existing results for the Dirichlet problem to second-order iterated Vekua equations and bicomplex equations, broadening the scope of solvable boundary value problems.

Findings

Proves well-posedness of the Dirichlet problem for iterated Vekua equations on non-circular conics.

Extends results for polyanalytic and generalized analytic functions to bicomplex differential equations.

Abstract

We prove the Dirichlet problem for second-order iterated Vekua equations, a natural generalization of the Bitsadze equation, is well-posed when the boundary condition is defined as a product of an exponential function and a polynomial on a non-degenerate conic that is not a circumference. Also, we extend this result, as well as other related results for the Dirichlet problem for polyanalytic and generalized analytic functions from the literature, to their analogues for bicomplex differential equations.