Diffraction at Corners for the Wave Equation on Differential Forms

Diffraction at Corners for the Wave Equation on Differential Forms

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 [23] and for the wave equation on systems ...

Geometric optics and the wave equation on manifolds with corners

According to geometric optics, light propagates in straight lines (in homogeneous media), reflects/refracts from surfaces according to Snell’s law: energy and tangential momentum are conserved. Thus, when reflecting from a hypersurface (which has codimension one) one gets the usual law of incident and reflected rays enclosing an equal angle to the normal to the surface. Indeed, conservation of ...

Asymptotics for the Wave Equation on Differential Forms on Kerr-de Sitter Space

We study asymptotics for solutions of Maxwell’s equations, in fact of the Hodge-de Rham equation (d+ δ)u = 0 without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which in particular include Schwarzschild-de Sitter spaces of all spacetime dimensions n ≥ 4. We prove that solutions decay expon...

The Pullback Equation for Differential Forms

Propagation of singularities for the wave equation on manifolds with corners

In this paper we describe the propagation of C∞ and Sobolev singularities for the wave equation on C∞ manifolds with corners M equipped with a Riemannian metric g. That is, for X = M × Rt, P = D t − ∆M , and u ∈ H loc (X) solving Pu = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WFb(u) is a union of maximally extended generalized broken bicharacteristics. This resul...