Wave Equation in Domains with Edges

Abstract

In this chapter, the wave equation is considered, at all times $t\in \mathbb {R}$, in a domain $G\subset \mathbb {R}^{n}$. The boundary ∂G contains finitely many smooth edges $\mathcal {M}_{m}$ of various dimensions $0\le {\mathrm {dim}}\mathcal {M}_{m}\le n-2$; outside of the union of edges, the boundary is smooth. The Dirichlet or Neumann conditions are given on $\partial G\backslash \cup _{m}\mathcal {M}_{m}$. We use this problem to demonstrate the method of combined estimates and derive the asymptotics of solutions near the edges.