A shape optimization problem for the first mixed Steklov–Dirichlet eigenvalue

A monotonicity result for the first Steklov–Dirichlet Laplacian eigenvalue

“…If n ≥ 3, they prove that σ ( ) achieves the maximum when the two balls are concentric. Subsequently, this result has been also proved for any dimension n ≥ 2 in [10], by using different proofs (see also [19] for an analogous result in two-points homogeneous spaces). Moreover, by performing numerical experiments, the authors in [14] exhibit that σ ( ) is monotone decreasing with respect to the distance between the centers of the two disks.…”

Where to place a spherical obstacle so as to maximize the first nonzero Steklov eigenvalue

“…Another interesting work in the same direction is due to Bonder, Groisman and Rossi, who studied the so called Sobolev trace inequality (see [9,20]), thus they were interested in the optimization of the first nonzero eigenvalue of an elliptic operator with mixed Dirichlet-Steklov boundary conditions among perforated domains: the existence and regularity of an optimal hole are proved in [22,23], and by using shape derivatives it is shown that annulus is a critical but not an optimum shape (see [22]). At last, we point out the recent papers [26,36,41,46,48], where the authors consider the first eigenvalue of the Laplace operator with mixed Dirichlet-Steklov boundary conditions. Many examples stated in the last paragraph deal with linear operators eigenvalues in the special case of doubly connected domains with spherical outer and inner boundaries.…”

Kuttler-Sigillito's Inequalities and Rellich-Christianson Identity

“…in Ω, u n = σu on F ⊂ ∂Ω, u = 0 on ∂Ω \ F, see e.g., [2,12,24]. When the Dirichlet conditions are not present, i.e., F = ∂Ω, (1.1) is the Steklov problem.…”