Propagation of singularities for the wave equation on manifolds with corners

Abstract

In this paper we describe the propagation of ${\mathcal C}^{\infty}$ and Sobolev singularities for the wave equation on ${\mathcal C}^{\infty}$ manifolds with corners $M$ equipped with a Riemannian metric $g$. That is, for $X=M\times\mathbb{R}_t$, $P=D_t^2-\Delta_M$, and $u\in H^1_{\mathrm{loc}}(X)$ solving $Pu=0$ with homogeneous Dirichlet or Neumann boundary conditions, we show that $\mathrm{WF}_{b}(u)$ is a union of maximally extended generalized broken bicharacteristics. This result is a ${\mathcal{C}}^{\infty}$ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if $M$ has a smooth boundary (and no corners).