On Eigenvalue Problems Related to the Laplacian in a Class of Doubly Connected Domains

Abstract

We study eigenvalue problems in some specific class of doubly connected domains. In particular, we prove the following.

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

“…It implies the kth mixed Steklov-Dirichlet eigenfunction is written by a product of a Laplacian eigenfunction and a radial function. By some computations as in Section 2.1 in [20], we can conclude that the kth mixed Steklov-Dirichlet eigenfunction is corresponding to the kth Laplacian eigenfunction. Since the first Laplacian eigenfunctions are constants, we obtain the following.…”

“…In this section, we derive an explicit formula for the first mixed Steklov-Dirichlet eigenfunctions in B 2 \ cl(B 1 ). Using the following standard argument as in [7] and [20], we can show that the first eigenfunction is a function that only depends on the distance from X.…”

“…The aim of this paper is to extend Verma's result [20] from Euclidean spaces to two-point homogeneous spaces. The main theorem is as follows.…”

“…Recently, Verma considered connected regions in R m with m ≥ 2 that are bounded by two spheres of given radii and gave the Dirichlet condition only on the inner sphere. Then the maximizer is obtained by the domain bounded by two concentric spheres [20].…”