R^n中迭代扰动Dirac方程双轴对称零解的Riemann-Hilbert问题
Riemann-Hilbert problems for bi-axially symmetric null-solutions to iterated perturbed Dirac equations in R^n
AI总结：
本文研究R^n双轴对称域上带Clifford代数值变系数的迭代扰动Dirac算子零解的Riemann-Hilbert边值问题，通过适配双轴对称的Almansi型分解与递归积分算子构造显式解，推广至扰动情形并推导Schwarz问题闭式解，建立高维高阶边值问题的统一框架。
摘要：
本文研究$\u211d^n$中双轴对称域上、具有Clifford代数值变系数的迭代扰动Dirac算子零解的Riemann-Hilbert边值问题（RHBVPs）。首先针对无扰情形的多元调和函数（即迭代Dirac算子的零解），通过适配双轴对称的Almansi型分解构造显式解，利用递归积分算子解耦层级结构。随后推广至矢量波数扰动的迭代Dirac算子，扩展该分解以处理谱各向异性同时保留对称性约束，确保Clifford代数参数化下的正则性。作为核心应用，推导了Schwarz问题的闭式解，展现了经典与高维情形下的统一结果。对称性、分解与摄动理论的结合为Clifford分析中的高阶边值难题建立了统一框架。
英文摘要：
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.
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