ON THE PO´ LYA-SCHIFFER CONVEXITY THEOREM AND ITS APPLICATIONS FOR EIGENVALUES OF VIBRATING STRINGS

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.