A R Aithal
Articles written in Proceedings – Mathematical Sciences
Volume 115 Issue 1 February 2005 pp 93-102
Volume 122 Issue 2 May 2012 pp 257-281
Let $\wp 1,\wp 0$ be two regular polygons of 𝑛 sides in a space form $M^2(\kappa)$ of constant curvature $\kappa=0,1$ 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 $\Omega =(\wp 1\backslash\wp 0)^0$, the interior of $\wp 1\backslash\wp 0$ in $M^2(\kappa)$. It is shown that the first Dirichlet’s eigenvalue $\lambda 1(\Omega)$ attains extremum when the axes of symmetry of $\wp 0$ coincide with those of $\wp 1$.
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