Riemann-Hilbert problems for bi-axially symmetric null-solutions to iterated perturbed Dirac equations in R^n
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913703458242560 |
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| author | Zuo, Dian Ku, Min He, Fuli |
| author_facet | Zuo, Dian Ku, Min He, Fuli |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16635 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Riemann-Hilbert problems for bi-axially symmetric null-solutions to iterated perturbed Dirac equations in R^n Zuo, Dian Ku, Min He, Fuli Analysis of PDEs Complex Variables 30G35, 15A66, 30E25, 35Q15, 31B10 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. |
| title | Riemann-Hilbert problems for bi-axially symmetric null-solutions to iterated perturbed Dirac equations in R^n |
| topic | Analysis of PDEs Complex Variables 30G35, 15A66, 30E25, 35Q15, 31B10 |
| url | https://arxiv.org/abs/2502.16635 |