Hyperbolic Systems in Domains with Conical Points

Abstract

In this chapter, a hyperbolic system of second-order differential equations

$$\displaystyle (\partial _{t}^{2}+\mathcal {P}(D_{x}))u(x,t)=f(x,t) $$

with Dirichlet or Neumann boundary conditions is considered, at all times $t\in \mathbb {R}$, in a wedge or a bounded domain with conical points on the boundary. The operator $\mathcal {P}(D_{x})$ is assumed to be formally self-adjoint and strongly elliptic. We study the asymptotics of solutions near an edge or conical points and deduce formulas for the coefficients in the asymptotics. The reasoning follows the scheme of Chap. 2 while the details of proofs become more complicated. In Sect. 3.1 we consider the Dirichlet problem in a wedge while Sect. 3.2 is devoted to the study of the Neumann problem in a cone and in a domain with a conical point.