[Paper Review] The Cauchy problem of the Lorentzian Dirac operator with APS boundary conditions

Nicolò Drago, Nadine Große+1 more|arXiv (Cornell University)|Apr 1, 2021

Spectral Theory in Mathematical Physics51 references4 citations

TL;DR

This paper establishes the well-posedness of the Cauchy initial-boundary value problem for the classical Lorentzian Dirac operator on globally hyperbolic spin manifolds with timelike boundary, under Atiyah-Patodi-Singer (APS) boundary conditions. By deriving energy estimates and employing mollifier-based regularization, the authors prove existence, uniqueness, and continuous dependence on data, extending the theory of Green operators to this non-elliptic, hyperbolic setting with non-local boundary conditions.

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.

Motivation & Objective

• To establish the well-posedness of the Cauchy problem for the Lorentzian Dirac operator on globally hyperbolic manifolds with timelike boundary.
• To extend the applicability of APS boundary conditions—previously used on spacelike boundaries—to timelike boundaries in a globally hyperbolic setting.
• To prove existence, uniqueness, and continuous dependence of weak solutions under non-local boundary conditions via energy estimates.
• To develop a regularization method using mollifiers to establish higher differentiability of solutions, requiring additional technical conditions.
• To construct Green operators for the Dirac operator with APS boundary conditions, proving the system is Green-hyperbolic.

Proposed method

• Derive energy estimates for the Dirac operator on globally hyperbolic manifolds with timelike boundary to ensure uniqueness and existence of weak solutions.
• Use a non-local mollifier construction adapted to the Dirac operator with self-adjoint boundary conditions, differing from classical local regularization techniques.
• Apply the open mapping theorem in Fréchet spaces to prove continuous dependence of solutions on initial and source data.
• Utilize a foliation structure $(M,g) acksimeq ([R\times\Sigma, -N^{2}dt^{2} + h_{t})$ to reduce the problem to a family of time-dependent Dirac operators on compact Riemannian manifolds with boundary.
• Define the solution space $\Gamma_{\textsc{aps}}(SM)$ as sections satisfying APS boundary conditions via spectral projection onto the negative eigenspace of the boundary Dirac operator $A_t$.
• Construct advanced and retarded Green operators $G_{\textsc{aps}}^{\pm}$ satisfying $D_M G_{\textsc{aps}}^{\pm}f = f$ and $G_{\textsc{aps}}^{\pm} D_M f = f$ for compactly supported sources.

Experimental results

Research questions

• RQ1 Can the Cauchy problem for the Lorentzian Dirac operator be well-posed when coupled to APS boundary conditions on a globally hyperbolic manifold with timelike boundary?
• RQ2 How can energy estimates be derived in the absence of ellipticity, given the hyperbolic nature of the Dirac operator?
• RQ3 Is it possible to achieve smoothness of solutions under non-local APS boundary conditions without relying on local coordinate patching?
• RQ4 Do Green operators exist for the Dirac operator with APS boundary conditions, and what are their causal properties?
• RQ5 How does the support of the solution propagate in time and space under these boundary conditions?

Key findings

• The Cauchy problem for the Lorentzian Dirac operator with APS boundary conditions is well-posed: a unique weak solution exists and depends continuously on the initial and source data.
• Energy estimates are derived for the Dirac operator on globally hyperbolic manifolds with timelike boundary, ensuring uniqueness and stability of solutions.
• Solutions are shown to be smooth under additional technical conditions, achieved via a novel mollifier-based regularization technique tailored to the Dirac operator with self-adjoint boundary conditions.
• The solution satisfies the causal propagation property: $\operatorname{supp}(\psi_T) \subseteq J(\operatorname{supp}(\psi_0) \cup \operatorname{supp}(f)) \cup J(\partial\Sigma)$, ensuring finite propagation speed.
• The existence of advanced and retarded Green operators $G_{\textsc{aps}}^{\pm}$ is proven, confirming that the system is Green-hyperbolic.
• The solution space $\Gamma_{\textsc{aps}}(SM)$ is invariant under time evolution, allowing the construction of a global solution from local solutions on $M_T$.