Sharp Trace Regularity for the Solutions of the Equations of Dynamic Elasticity

Abstract

Sharp trace regularity results are obtained for the system of linear elasticity, relating the norm of the tangential derivative of the solution on the boundary to the norm of the time derivative. To prove this relationship, we take advantage of the techniques of pseudodifferential calculus and microlocal analysis. Via a partition of unity, a smoothing of the boundary procedure, and a change of variable, the system can be shown to be equivalent to a more general problem on a half-space. This transition to a half-space problem facilitates localization of the solution and, thus, its behavior, near the boundary. By studying the symbol of the resulting differential operators after the problem has been converted to a half-space, a bound of the tangential derivative in terms of the time derivative is obtained. Key words: linear elasticity, trace regularity, microlocal analysis AMS Subject Classifications: 35B65, 73C02 1 Introduction Sharp trace regularity results have proven themselves to ..