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Mathematics
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Clifford Analysis: Theory, Methods, and Multidisciplinary Applications
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Clifford Analysis: Theory, Methods, and Multidisciplinary Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C: Mathematical Analysis".

Deadline for manuscript submissions: 31 March 2027 | Viewed by 2065

Editor

Dr. Ji-eun Kim

E-Mail Website
Guest Editor
Division of Education, Dongguk University, Gyeongju 38066, Republic of Korea
Interests: quaternion analysis; mathematical physics; computational methods; partial differential equations; several complex analysis

Special Issue Information

Dear Colleagues,

This Special Issue, “Clifford Analysis: Theory, Methods, and Multidisciplinary Applications”, offers a curated selection of pioneering research situated at the intersection of algebra, analysis, and geometry. Clifford analysis, as an extension of complex analysis into higher dimensions, provides powerful algebraic and analytical tools for exploring monogenic functions, Dirac-type operators, and the concept of conformal invariance. This Special Issue aims to deepen our understanding of the theoretical foundations of Clifford analysis while showcasing its growing relevance across a range of disciplines.

We welcome contributions that address both the theoretical development and the applied aspects of the field. Topics of interest include, but are not limited to, advancements in partial differential equations, hypercomplex function theory, geometric calculus, and their applications in mathematical physics, computer vision, and signal/image processing. By integrating foundational insights with practical methodologies, this Special Issue demonstrates how Clifford algebras provide a cohesive framework that effectively connects the domains of pure and applied mathematics. t is intended as a resource for both specialists in geometric analysis and those exploring novel applications of Clifford theory in broader scientific contexts.

Dr. Ji-eun Kim
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • clifford analysis
  • monogenic functions
  • dirac operator
  • hypercomplex analysis
  • partial differential equations
  • geometric calculus
  • mathematical physics
  • signal and image processing

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Published Papers (2 papers)

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Research

24 pages, 413 KB  
Article
Biregular Mappings on H×H: Domains of Hyperholomorphy, Integral Representations, and Runge Approximation
by Ji Eun Kim
Mathematics 2026, 14(4), 682; https://doi.org/10.3390/math14040682 - 14 Feb 2026
Viewed by 620
Abstract
We develop a PDE and boundary integral framework for quaternion-valued fields on product domains Ω⊂H×H governed by the mixed left/right Cauchy–Fueter system We identify the natural compatibility condition and prove local solvability with quantitative H1 estimates, as well [...] Read more.
We develop a PDE and boundary integral framework for quaternion-valued fields on product domains Ω⊂H×H governed by the mixed left/right Cauchy–Fueter system We identify the natural compatibility condition and prove local solvability with quantitative H1 estimates, as well as global weak solvability on admissible products Ux×Uy. Motivated by these estimates, we introduce domains of hyperholomorphy and hyper-conjugates for data that are harmonic in each factor (Δxu=Δyu=0), and we establish Carleman-type quantitative unique continuation tools (boundary blow-up, three-balls, and doubling), including a propagation-of-smallness principle across the two factors. On the potential-theoretic side, we construct a double boundary integral representation for biregular fields with kernel K(ξ,η;x,y)=E(ξ−x)E(y−η), establish mapping and jump relations for the associated layer potentials on Lipschitz boundaries, and obtain a Fredholm boundary integral equation for the boundary density in the smooth admissible regime. Finally, we prove a constructive Runge approximation theorem on admissible products and outline a practical discretization workflow consistent with the analysis. Full article
(This article belongs to the Special Issue Clifford Analysis: Theory, Methods, and Multidisciplinary Applications)
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17 pages, 340 KB  
Article
O-Regular Mappings on C(C): A Structured Operator–Theoretic Framework
by Ji Eun Kim
Mathematics 2025, 13(20), 3328; https://doi.org/10.3390/math13203328 - 18 Oct 2025
Viewed by 839
Abstract
Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural [...] Read more.
Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural CR systems, a canonical first derivative, and a Cauchy-type boundary identity on the commutative model C(C)≅C4 has not been systematically developed. Purpose and Aims. This paper develops such a calculus for O-regular mappings on C(C) and establishes three pillars of the theory. Main Results. (i) A fully coupled Cauchy–Riemann system characterizing O-regularity; (ii) identification of a canonical first derivative g′(z)=∂x0g(z); and (iii) a Stokes-driven boundary annihilation law ∫∂Ωτg=0 for a canonical 7-form τ. On (pseudo)convex domains, ∂¯-methods yield solvability under natural compatibility and regularity assumptions. Stability (under algebra-preserving maps), Liouville-type, and removability results are also obtained, and function spaces suited to this algebra are outlined. Significance. The results show that a large portion of the classical holomorphic toolkit survives, in algebra-aware form, on C(C). Full article
(This article belongs to the Special Issue Clifford Analysis: Theory, Methods, and Multidisciplinary Applications)
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