Sharp Trace Regularity for the Solutions of the Equations of Dynamic Elasticity
نویسنده
چکیده
Sharp trace regularity results have proven themselves to be of critical importance in the study of controllability and stabilizability of various systems, as well as being of great interest in their own right. Particular cases include the wave equation (see [9]) and both linear and nonlinear plate equations (see [5], [8]). In our study of the three-dimensional system of linear elasticity, we focus on results for the wave equation. This is due to the fact that, under appropriate assumptions, the system of elasticity can be decoupled into three wave equations. Thus, we would hope that results, analagous to those available for the wave equation, would hold in the fully coupled case. Our motivation in developing these trace estimates arises from the desire to eliminate the strong geometric constraints assumed to hold in most results on boundary stabilization for the system of elasticity (see e.g. [7, 6]). In the case of the wave equation, stabilization results are numerous. However, until the works of Lasiecka and Triggiani [9] and Bardos, Lebeau and Rauch [2], most results were based on the assumption that the geometry of the domain satis ed strict constraints. A critical step in removing these constraints in [9] was a pseudodi erential analysis which permits certain boundary traces of the solution to the wave equation to be expressed in terms of other traces modulo lower-order interior terms. Estimates of solutions near the boundary have a long history, dating back to such works as that of Agmon, Douglis and Nirenberg [1]. In what follows, we will focus on the proof of trace regularity, while the question of stabilization without geometric constraints will be addressed in a subsequent paper. To formulate the system of elasticity, we begin with the following de nitions. Let u = (ui), 1 i n be the displacement vector. Since we are considering a
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