Riemann-Hilbert problems for bi-axially symmetric null-solutions to iterated perturbed Dirac equations in R^n

TL;DR

This paper develops explicit solutions for Riemann-Hilbert boundary value problems involving iterated perturbed Dirac operators in bi-axially symmetric domains, extending Clifford analysis techniques to handle spectral anisotropy and perturbations.

Contribution

It introduces a bi-axially adapted Almansi-type decomposition for null solutions to iterated Dirac operators, generalizes to perturbed cases with spectral anisotropy, and derives closed-form solutions for the Schwarz problem.

Findings

Explicit solutions for unperturbed poly-monogenic functions.

Extension of decomposition to perturbed Dirac operators with spectral anisotropy.

Closed-form solutions to the Schwarz boundary value problem.

Abstract

This work addresses Riemann-Hilbert boundary value problems (RHBVPs) for null solutions to iterated perturbed Dirac operators over bi-axially symmetric domains in $\mathbb{R}^n$ with Clifford-algebra-valued variable coefficients. We first resolve the unperturbed case of poly-monogenic functions, i.e., null solutions to iterated Dirac operators, by constructing explicit solutions via a bi-axially adapted Almansi-type decomposition, decoupling hierarchical structures through recursive integral operators. Then, generalizing to vector wave number-perturbed iterated Dirac operators, we extend the decomposition to manage spectral anisotropy while preserving symmetry constraints, ensuring regularity under Clifford-algebraic parameterizations. As a key application, closed-form solutions to the Schwarz problem are derived, demonstrating unified results across classical and higher-dimensional settings. The interplay of symmetry, decomposition, and perturbation theory establishes a cohesive framework for higher-order boundary value challenges in Clifford analysis.