Project Details
Projekt
Geometric operators on singular domains
Subject Area
Mathematics
Term
from 2017 to 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 338892245
Boundary value problems are and were extensively studied due to their applications to physics, geometry, and numerical analysis. If the boundary is smooth, we have a very good understanding of the standard Dirichlet and Neumann boundary conditions. Here, the regularity of the solutions is only obstructed by the one of the boundary values and the image under the operator under consideration. For boundaries with singularities, in particular in higher dimensions, the problem is not completely well-understood. In this project we study the question of well-posedness for the Laplacian with mixed boundary conditions on singular domains. We stress that regularity and well-posedness results are also important for numerical applications. For example the order of convergence of usual Galerkin schemes are determined by the Sobolev scale. The underlying idea used for our programme is that a conformal change to a metric by an appropriate weight function can transform domains with stratified boundary into a manifold of bounded geometry with boundary by sending the singularities of the boundary to infinity. While the well-posedness results on the singular domains are always in some weighted Sobolev space we can treat with a conformal blow-up the boundary value problem obtained by translation to the noncompact blow-up with standard Sobolev spaces. This will allow for a uniform treatment of more general singularities also in higher dimensions.In the second part of our project we study the nonrelativistic Schrödinger operator for N electrons in a Coulomb-type potential. In particular we are interested in the regularity of the eigenfunctions. This is of high relevance to applications in physics and chemistry as it helps to establish improved adaptive algorithms for numerical calculations of the eigenfunctions. Although this is different from the well-posedness question from above, similar ideas will be applied. We treat the singularities of the potential, i.e. points of multi-electron or electron-nucleus collisions, as singularities of the boundary value problems which we blow-up again. This will be combined with existing tools like natural compactifications. We also incorporate the Kustaanheimo-Stiefel transform in our picture, a method which was previously successfully applied for Coulomb potentials in classical mechanics and for strong regularity results for eigenfunctions of the Schrödinger operator in two-particle collisions. Also for Schrödinger eigenfunctions, our results may serve as a starting point for numerical algorithms in the future.
DFG Programme
Priority Programmes
Subproject of
SPP 2026:
Geometry at Infinity
