Boundary Value Problems for Clifford-Analytic Operators on Singular Domains

About the Project

These projects are open to students worldwide, but have no funding attached. Therefore, the successful applicant will be expected to fund tuition fees at the relevant level (home or international) and any applicable additional research costs. Please consider this before applying.

This PhD project is at the intersection of complex analysis, partial differential equations, and geometric analysis. It aims to extend recent results for the Cauchy-Riemann operator on corner domains to higher dimensions, within the framework of Clifford analysis. The Cauchy–Riemann equations, central to complex analysis, admit natural generalizations to higher-dimensional space via Dirac-type operators, which are a key concept in Clifford analysis, a powerful higher-dimensional analogue of holomorphic function theory.

The project focuses on boundary value problems for these operators on polygonal and polyhedral domains, where the presence of corners and edges leads to subtle analytical difficulties. In smooth domains, the theory of such operators is well-understood; however, the singular geometry of non-smooth boundaries introduces new phenomena such as loss of regularity, corner singularities, and delicate compatibility conditions for boundary data. The goal is to develop a robust framework to describe the mapping properties, parametrices, and Green-type formulas for Clifford-analytic operators on such singular domains - generalizing results previously obtained for the 2-dimensional Cauchy-Riemann operator.

The successful applicant will:

• Develop a deep understanding of linear PDE theory, particularly elliptic operators and boundary value problems.
• Study Green’s functions, Green’s formulas, and parametrix constructions on domains with corners and edges.
• Explore connections between functional analysis, potential theory, and Clifford algebras.

This is a strongly analytic project: candidates should already have a solid background in real and complex analysis, including familiarity with topics such as Cauchy’s theorem, contour integration, and harmonic functions. During the PhD, the student will build expertise in advanced analysis, Sobolev spaces, singular integral operators, and geometric aspects of PDEs. A basic knowledge of differential equations and functional analysis will be helpful.

Informal enquiries can be made by contacting Dr M Upmeier (markus.upmeier@abdn.ac.uk).

Decisions will be based on academic merit. The successful applicant should have, or expect to obtain, a UK Honours Degree at 2.1 (or equivalent) in Mathematics.

We encourage applications from all backgrounds and communities, and are committed to having a diverse, inclusive team.

Application Procedure:

Formal applications can be completed online: https://www.abdn.ac.uk/pgap/login.php.

You should apply for Degree of Doctor of Philosophy in Mathematics to ensure your application is passed to the correct team for processing.

Please clearly note the name of the lead supervisor and project titleon the application form. If you do not include these details, it may not be considered for the project.

Your application must include: A personal statement, an up-to-date copy of your academic CV, and clear copies of your educational certificates and transcripts.

Please note: you do not need to provide a research proposal with this application.

If you require any additional assistance in submitting your application or have any queries about the application process, please don't hesitate to contact us at researchadmissions@abdn.ac.uk

Funding Notes

This is a self-funding project open to students worldwide. Our typical start dates for this programme are February or October.

Fees for this programme can be found here Finance and Funding | Study Here | The University of Aberdeen.

References

1. P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, 2011.
2. J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Monographs in Mathematics, 2012.
3. F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Research Notes in Mathematics, Pitman, 1982.