On the extrema of Dirichlet’s first eigenvalue of a family of punctured regular polygons in two dimensional space forms

Abstract

Let \( \wp_{1}, \wp_0 \) be two regular polygons of n sides in a space form M 2(κ) of constant curvature κ = 0,1 or − 1 such that \(\wp_0 \subset \wp_1\) and having the same center of mass. Suppose \( \wp_{0} \) is circumscribed by a circle C contained in \( \wp_{1} .\) We fix \(\wp_1\) and vary \(\wp_0\) by rotating it in C about its center of mass. Put \(\Omega=(\wp_{1} \setminus \wp_{0})^{0},\) the interior of \( \wp_{1} \setminus \wp_{0} \) in M 2(κ). It is shown that the first Dirichlet’s eigenvalue λ 1( Ω) attains extremum when the axes of symmetry of \( \wp_{0} \) coincide with those of \( \wp_{1}.\)