The well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic manifolds with timelike boundary

TL;DR

This paper proves the well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic manifolds with timelike boundary, using boundary conditions and hyperbolic system techniques.

Contribution

It establishes existence, uniqueness, and smoothness of solutions for the Dirac operator with MIT-boundary conditions on such manifolds, extending to more general boundary conditions.

Findings

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

Demonstrated existence and uniqueness of weak solutions.

Extended results to a broader class of boundary conditions.

Abstract

We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then using local methods developed by Lax, Phillips and Rauch, Massey to show smoothness of the solutions. Our proof actually works for a slightly more general class of local boundary conditions.