The Cauchy problem of the Lorentzian Dirac operator with APS boundary conditions

TL;DR

This paper establishes well-posedness and regularity results for the Lorentzian Dirac operator with APS boundary conditions on globally hyperbolic manifolds, using energy estimates and mollifier techniques.

Contribution

It provides the first rigorous analysis of the Dirac operator with APS boundary conditions in a Lorentzian setting, including existence, uniqueness, and differentiability of solutions.

Findings

Proved well-posedness of the Cauchy problem with APS boundary conditions.

Derived energy estimates crucial for solution analysis.

Studied solution regularity using mollifier operators.

Abstract

We consider the classical Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to APS-boundary conditions. This is achieved by deriving suitable energy estimates, which play a fundamental role in establishing uniqueness and existence of weak solutions. Finally, by introducing suitable mollifier operators, we study the differentiability of the solutions. For obtaining smoothness we need additional technical conditions.