Inserted: 28 oct 2020
Last Updated: 24 jul 2021
Journal: Journal of Differential Equations
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. Together with the shape calculus, these fine geometric properties help us to prove that the first eigenvalue is strictly decreasing as the inner ball moves towards the boundary of the outer ball.
Keywords: shape derivative, Torsional rigidity, Geometry of first eigenfunction, Foliated Schwarz symmetry, Monotonicity of first eigenvalue, Zaremba problem