Non-local boundary conditions - for example the Atiyah-Patodi-Singer (APS) conditions - for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, Bär-Strohmaier and Drago-Große-Murro introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While Bär--Strohmaier showed the Fredholmness of the Dirac operator with these boundary conditions, Drago-Große-Murro proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions.
In this talk, I will follow in the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still well-posed.
After a brief introduction to globally hyperbolic manifolds with timelike boundary and the Dirac operator on these manifolds, I will discuss non-local boundary conditions and the related Cauchy problems in this setting. In the last part of my talk I will introduce important examples of non-local boundary conditions that lead to well-posedness of the corresponding Cauchy problem.