  We discuss some of the geometric properties, such as the foliated Schwarz
symmetry, the monotonicity along the axial and the affine-radial directions, of
the first eigenfunctions of the Zaremba problem for the Laplace operator on
annular domains. These fine geometric properties, together with the shape
calculus, help us to prove that the first eigenvalue is strictly decreasing as
the inner ball moves towards the boundary of the outer ball.
