Convergence rate for eigenvalues of the elastic Neumann--Poincare operator on smooth and real analytic boundaries in two dimensions

The elastic Neumann--Poincare operator is a boundary integral operator associated with the Lame system of linear elasticity. It is known that if the boundary of a planar domain is smooth enough, it has eigenvalues converging to two different points determined by Lame parameters. We show that eigenvalues converge at a polynomial rate on smooth boundaries and the convergence rate is determined by smoothness of the boundary. We also show that they converge at an exponential rate if the boundary of the domain is real analytic.