The paper deals with Dirichlet and Neumann problems for the Lamé operator. In the exterior \(\Omega= \mathbb{R}^3\backslash\omega\) of a bounded domain, these problems are associated with the family of boundary value problems in domains \(\Omega_R= \{x\in\Omega:|x|< R\}\) with diameter \(2/R\), where \(R\) is a large parameter. On the external part of the boundary \(\partial\Omega_R\) one of the three boundary conditions: Dirichlet, Neumann, or mixed is posed. The asymptotics are found and estimates for the solutions are derived in weight classes \(L_p\) being asymptotically exact as \(R\to\infty\).