Riemann-Hilbert problems for bi-axially symmetric null-solutions to iterated perturbed Dirac equations in R^n
Dian Zuo, Min Ku, Fuli He
TL;DR
This work develops a Clifford-algebraic framework for Riemann-Hilbert boundary value problems governing bi-axially symmetric null solutions to iterated Dirac equations in $\mathbb{R}^n$. Central to the approach is a bi-axially adapted Almansi-type decomposition that expresses solutions as $\varphi(\underline{x})=e^{\beta}\sum_{j=0}^{k-1} \underline{x}^j \varphi_j(\underline{x})$ with $\beta=\sum_{j=1}^n \alpha_j x_j$ and monogenic components $\varphi_j$, enabling explicit integral representations via operator tools $I_s$ and $\mathcal{Q}_k$. The paper first resolves the unperturbed poly-monogenic case and then extends to vector-wave-number perturbed systems, preserving bi-axial symmetry and yielding Schwarz-type closed-form solutions along with robust existence-uniqueness results for Hölder and $L_p$ data. The work integrates Fueter-type ideas into Clifford calculus to address higher-order, anisotropic boundary-value problems, offering a unified method with potential applications in anisotropic media, photonic crystals, and quantum-elastic contexts.
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.