Abstract
\noindent We consider eigenvalue problems for the vibrating string \vspace{-0.1cm} \[ u^{\prime \prime }(x)+\lambda \rho (x)u(x)=0,\;\;\;\;u(0)=u(a)=0 \] \vspace{-0.6cm} \noindent where the density $\rho (x)\;$is a positive continuous function on $% [0,a]$. Let $\lambda _{n}(t)$ be the $n$th eigenvalue of the string with $\rho =\rho (x,t)$. A classical convexity theorem of P\'{o}lya and Schiffer states that for any $k\geq 1$, the sum $% \sum_{n=1}^{k}\frac{1}{\lambda _{n}(t)}\;$is a convex function of $t$ if $\rho (x,t)$ is convex with respect to $t$. In this paper, we shall give a different approach to this result based on variational analysis. The ideas used also lead to applications in the case of symmetric densities and in the case of concave densities.
Citation
Min-Jei Huang. "ON THE PO´ LYA-SCHIFFER CONVEXITY THEOREM AND ITS APPLICATIONS FOR EIGENVALUES OF VIBRATING STRINGS." Taiwanese J. Math. 8 (3) 489 - 497, 2004. https://doi.org/10.11650/twjm/1500407668
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