= Radial diffusion Rayleigh quotient
{title2=$\mathcal R[R]=\frac{\int_0^1r\kappa(r)R'^2\,dr}{\int_0^1rR^2\,dr}$}
For axisymmetric <heat conduction> in a unit disk, separated modes satisfy $-(r\kappa R')'=\nu rR$, with weight $r$, outer <Dirichlet boundary condition> $R(1)=0$, and zero radial flux at the origin. <Integration by parts> gives the displayed <Rayleigh quotient>. Its <Euler-Lagrange equation> under fixed $\int rR^2$ recovers the mode equation; the natural endpoint term is $r\kappa R'\delta R$. The <Rayleigh-Ritz variational principle> says that every admissible nonzero trial profile gives an upper bound on the bottom of the <spectrum>, and on the lowest <eigenvalue> when this bottom is attained by a <fundamental diffusion mode>. Finite weighted energy, rather than pointwise existence of a second derivative at the origin, is the relevant trial admissibility requirement.
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