Incompressible limit of the Hookean elastodynamics in a bounded domain

Previously, the incompressible limit of the equations of Hookean elastodynamics has been established in the whole space. The present paper is devoted to the study of the incompressible limit in a bounded domain. Here, the main difficulty lies in the interactions between the large operator and the boundary. By designing some delicate semi-norms and energy estimates, we obtain the uniform estimates of the solutions. As a result, the incompressible limit is established by classical compactness argument.