= Power-law variational bound for a circular membrane
{title2=$\nu_1\leq c(1+\sqrt2)/(2\pi a)$}
For a fixed circular <elastic membrane>, the <Rayleigh-Ritz variational principle> applied to $f(r)=1-(r/a)^p$, $p>0$, gives $\lambda_1\leq(p+3+2/p)/a^2$. The numerator of its <Rayleigh quotient> is $p/2$, and the denominator is $a^2p^2/[2(p+1)(p+2)]$. Minimization gives $p=\sqrt2$, $\lambda_1\leq(3+2\sqrt2)/a^2$, and the displayed upper bound for the ordinary <frequency> $\nu_1=c\sqrt{\lambda_1}/(2\pi)$. Every $p>0$ has finite radial Dirichlet energy, which is the required variational admissibility condition.
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