• On the Extrema of Dirichlet's First Eigenvalue of a Family of Punctured Regular Polygons in Two Dimensional Space Forms

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    • Keywords

       

      Laplace Beltrami operator; extremum of first eigenvalues; Green’s identities for polygonal domains; Hadamard formula for polygonal domains.

    • Abstract

       

      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 $\wp 1$.

    • Author Affiliations

       

      A R Aithal1 2 Rajesh Raut1 2

      1. Department of Mathematics, University of Mumbai, Vidyanagari, Mumbai 400 098, India
      2. R. D. National College & W. A. Science College, Bandra, Mumbai 400 050, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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