Dirac operators on polygonal and polyhedral domains

• Applicant: Professorin Dr. Nadine Große
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 560457657

Dirac operators play e.g. a role in the description of massive relativistic particles of spin 1/2 and graphene, but also in geometric questions as the study of manifolds admitting metrics of positive scalar curvature. As for all differential operators boundary value problems arise in several applications. Boundary value problems for the Dirac operators for smooth boundaries are studied since quite some time. Two motivations to study these comes from the description of particles confined to certain regions and from the geometric side from the Atiyah-Patodi-Singer index theorem. Nevertheless, first-order boundary value problems are technically more involved than the ones for second order operators. A comprehensive treatment of first-order boundary value problems for Dirac-type operator from a functional analytic point of view was given by Bär and Ballmann around 2010. In this proposal boundary value problems for the Dirac operator on singular domains shall be studied -- mainly aiming for polyhedra as domain. Why? First, thinking of numerical simulations it is rather natural to consider such domains since these occur when the domain is meshed. Secondly, it is mathematically interesting on its own, since the methods used are not just generalizations of the one used for smooth boundaries. There are already several works for curvilinear polygons, sectors or convex cones in 3D - mainly concerning questions like self-adjointness and spectral properties and under the most common local boundary conditions. The main aim of this proposal is to extend these works to more general singular domains and more local boundary conditions and also include the question of well-posedness. The latter question is relevant to obtain regularity estimates that can be used to prove e.g. rate of convergence in numerical algorithms.

• DFG Programme: Research Grants