On the first Steklov-Dirichlet eigenvalue for eccentric annuli

In this paper, we investigate the first Steklov-Dirichlet eigenvalue on eccentric annuli. The main geometric parameter is the distance t between the centers of the inner and outer boundaries of an annulus. We first show the differentiability of the eigenvalue in t and obtain an integral expression for the derivative value in two and higher dimensions. We then derive an upper bound of the eigenvalue for each t, in two dimensions, by the variational formulation. We also obtain a lower bound of the eigenvalue, given a restriction that the two boundaries of the annulus are sufficiently close. The key point of the proof of the lower bound is in analyzing the limit behavior of an infinite series expansion of the first eigenfunction in bipolar coordinates. We also derive a relation between the first eigenvalue and a sequence of eigenvalues obtained by a finite section method. Based on this relation, we also perform numerical experiments that exhibit the monotonicity for two dimensions.