On eigenvalue problems related to the laplacian in a class of doubly connected domains

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

“…In [48], the authors consider a mixed Dirichlet-Steklov eigenvalue problem. They prove that the first nonzero eigenvalue is maximal when the balls are concentric in dimensions larger or equal than 3 (see [48,Theorem 1]) and remark that the planar case remains open (see [48,Remark 2]). We show that the ideas developed in this paper allow us to give an alternative and simpler proof of [48,Theorem 1].…”

“…They prove that the first nonzero eigenvalue is maximal when the balls are concentric in dimensions larger or equal than 3 (see [48,Theorem 1]) and remark that the planar case remains open (see [48,Remark 2]). We show that the ideas developed in this paper allow us to give an alternative and simpler proof of [48,Theorem 1]. Then we extend this result to the planar case.…”

“…In this section, we show that the ideas of our proof in Section 2 also apply to the problem considered in [48]. Thus, we give an alternative proof of [48, Theorem 1] which deals with n ≥ 3 and tackle the planar case which is to our knowledge still open (see [48,Remark 2]).…”

“…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.…”

“…Unfortunately, this is not the case for the inequality on the boundary (assertion 3) for which the author needs more computations (see [48,Section 2.2]).…”