Diffraction at corners for the wave equation on differential forms

TL;DR

This paper establishes the propagation of singularities for the wave equation on differential forms with natural boundary conditions on Lorentzian manifolds with corners, extending previous scalar results to more complex boundary conditions.

Contribution

It proves propagation of singularities for the wave equation on differential forms with natural boundary conditions on manifolds with corners, a novel extension of prior scalar wave results.

Findings

Propagation of singularities established for differential forms

Results include Maxwell's equations formulation

Addresses boundary conditions at corners of codimension ≥ 2

Abstract

In this paper we prove the propagation of singularities for the wave equation on differential forms with natural (i.e. relative or absolute) boundary conditions on Lorentzian manifolds with corners, which in particular includes a formulation of Maxwell's equations. These results are analogous to those obtained by the author for the scalar wave equation and for the wave equation on systems with Dirichlet or Neumann boundary conditions earlier. The main novelty is thus the presence of natural boundary conditions, which effectively make the problem non-scalar, even `to leading order', at corners of codimension at least 2.