Elastodynamics in domains with edges

Abstract:

Time-dependent boundary value problems with given displacements or stresses on the boundary of a domain are considered. The purpose is to describe the asymptotics of solutions near the edges of the boundary (including formulas for the “stress intensity factors”). The approach is based on various (energy and weighted) estimates of solutions. The weighted estimates in question are mixed in the sense that, in distinct zones, they involve derivatives of different orders. The method is implemented for problems in the cylinder $\mathbb D \times \mathbb R$, where $\mathbb D$ is an $m$-dimensional wedge, $m\geq 2$, and $\mathbb R$ is the time axis. For the cylinder $G\times \mathbb R$, where $G$ is a bounded domain with edges on the boundary, all the steps of the method are described except for the final one, which is related to the asymptotics itself. This step consists in compiling some known results of the theory of elliptic boundary value problems.