Generalizations of the Dirichlet problem for bianalytic functions

Technical Summary: Generalizations of the Dirichlet Problem for Bianalytic Functions

Problem Statement
The paper addresses the well-posedness of the Dirichlet boundary value problem for higher-order iterated Vekua equations, specifically focusing on second-order equations that generalize the Bitsadze equation ($\frac{\partial^2 w}{\partial \bar{z}^2} = 0$). While the classical Dirichlet problem for harmonic functions is well-posed on planar domains, the problem for the Bitsadze equation on the complex unit disk is known to be ill-posed, admitting infinitely many solutions for zero boundary data. Recent literature (specifically Moreno García et al., [26]) established that the Bitsadze Dirichlet problem becomes well-posed when the boundary is a non-degenerate conic that is not a circumference and the boundary condition is a polynomial. This work seeks to extend these findings to two broader contexts:

1. Second-order iterated Vekua equations: Determining if the well-posedness on non-circular conics extends to equations of the form $(\frac{\partial}{\partial \bar{z}} - A)^2 w = 0$, where $A$ is a coefficient function.
2. Bicomplex extensions: Generalizing results for complex-valued functions to bicomplex-valued functions (functions taking values in the four-dimensional algebra of bicomplex numbers $\mathbb{B}$), which are relevant for studying the complex stationary Schrödinger equation.

Methodology
The author employs the idempotent representation of bicomplex numbers to decompose bicomplex-valued problems into pairs of classical complex-valued problems.

• Idempotent Decomposition: Any bicomplex number $w$ can be uniquely written as $w = p_+ w_+ + p_- w_-$, where $p_\pm$ are idempotent elements. This allows the bicomplex differential operators (such as $\bar{\partial}$ and the iterated Vekua operator) to be expressed as linear combinations of standard complex differential operators acting on the components $w_+$ and $w_-$.
• Similarity Principle: The paper utilizes the "similarity principle" (Theorem 2.1), which states that solutions to Vekua equations can be represented as the product of an exponential function (determined by the coefficient $A$ via the Vekua integral operator $T_D$) and a holomorphic (or generalized analytic) function.
• Reduction Strategy: The core methodology involves reducing the bicomplex Dirichlet problem to two independent complex Dirichlet problems. By solving the complex components and recombining them via the idempotent basis, the author constructs solutions for the bicomplex system. This approach is applied to both the unit disk and non-degenerate conic boundaries.

Key Contributions and Results

1. Well-posedness on Non-Circular Conics for Vekua-Bitsadze Equations:
   The paper proves that the Dirichlet problem for the second-order iterated Vekua equation $(\frac{\partial}{\partial \bar{z}} - A)^2 w = 0$ is well-posed on a domain $D$ enclosed by a non-degenerate conic $\Gamma$ (that is not a circumference), provided the boundary condition is of the form $w = e^{T_D[A]}P$, where $P$ is a polynomial and $T_D[A]$ is the Vekua integral operator applied to the coefficient $A$. This generalizes the result of Moreno García et al. [26] from the Bitsadze equation ($A \equiv 0$) to the general Vekua-Bitsadze case.

2. Bicomplex Extension of Ill-posedness on the Unit Disk:
   The author demonstrates that the ill-posedness of the Dirichlet problem on the unit disk persists in the bicomplex setting. Specifically, the bicomplex Bitsadze equation ($\bar{\partial}^2 w = 0$) and the bicomplex iterated Vekua equation ($( \bar{\partial} - A)^n w = 0$) admit infinitely many linearly independent solutions when the boundary condition is zero on the unit circle. This is established by constructing solutions from the infinite families of solutions available in the complex components.

3. Characterization of Solvability on the Disk:
   Necessary and sufficient conditions for the unique solvability of the Dirichlet problem for bicomplex higher-order iterated Vekua equations on the unit disk are derived. These conditions are expressed as integral constraints on the boundary data and the non-homogeneous part, mirroring the conditions found in the complex case (Theorem 2.19). The solution is explicitly represented as a sum of integrals involving the boundary data and the bicomplex Vekua operator.

4. Bicomplex Well-posedness on Non-Circular Conics:
   The paper establishes that the Dirichlet problem for the bicomplex Bitsadze equation and the bicomplex Vekua-Bitsadze equation is well-posed when the boundary is a non-degenerate conic (not a circumference). The boundary condition for the Vekua-Bitsadze case is shown to be the product of a polynomial and a bicomplex exponential function determined by the coefficient $A$ via the bicomplex integral operator $T_D^B$.

Significance
The paper claims to provide a complete characterization of the solvability of the Dirichlet problem for arbitrarily higher-order Vekua equations in terms of the solvability of two associated complex Dirichlet problems. By leveraging the idempotent representation, the work unifies the theory of complex generalized analytic functions with bicomplex analysis. It confirms that the structural properties governing well-posedness (specifically the dependence on the geometry of the boundary and the form of the boundary data) observed in the complex case for the Bitsadze equation hold true for their bicomplex analogues and for higher-order iterated Vekua equations. The results extend the known theory of boundary value problems from the complex plane to the bicomplex setting, offering a framework for analyzing bicomplex differential equations that arise in physical modeling, such as the stationary Schrödinger equation.