Extend solvability/uniqueness theory to non-Friedrichs boundary conditions

Develop a solvability, uniqueness, and regularity theory for wave-type equations on Lorentzian manifolds with timelike curves of cone-type singularities under boundary conditions other than those arising from the Friedrichs extension, for example the natural boundary conditions for differential forms considered by Vasy (2010, "Diffraction at corners for the wave equation on differential forms"). Establish forward well-posedness and microlocal propagation results analogous to those proved for the Friedrichs extension in the symmetric ultrastatic setting.

Background

The paper develops a general microlocal framework for existence, uniqueness, and regularity of solutions to wave-type equations on Lorentzian manifolds featuring timelike curves of conic singularities. In symmetric ultrastatic settings, the results recover solutions produced by the functional calculus for the Friedrichs extension.

However, the analysis is currently tied to the Friedrichs extension when comparing with classical spectral theory in the ultrastatic case. Boundary value problems with different boundary conditions, such as those arising for differential forms (e.g., absolute/relative conditions) as treated by Vasy (2010), fall outside the present method.

Extending the theory to these non-Friedrichs boundary conditions would broaden its applicability, for instance to wave equations on forms and systems where natural boundary conditions differ from the self-adjoint Friedrichs realization.