= Solution
For a nonzero admissible radial profile define the <radial diffusion Rayleigh quotient>
$$
\mathcal R[R]=\frac{\int_0^1r\kappa R'^2\,dr}{\int_0^1rR^2\,dr}.
$$
Minimizing the numerator subject to $\int_0^1rR^2\,dr=1$ introduces a multiplier $\nu$. The <Euler-Lagrange equation> for $r\kappa R'^2-\nu rR^2$ is
$$
(r\kappa R')'+\nu rR=0.
$$
The allowed variations vanish at $r=1$, and the natural condition at the centre is zero radial flux. Thus the minimum, when attained, is the smallest decay <eigenvalue> $\nu_0$. Equivalently, expansion in weighted-orthogonal <eigenfunctions> makes $\mathcal R[R]$ a weighted average of their decay <eigenvalues>. The <Rayleigh-Ritz variational principle> therefore gives
$$
\boxed{\nu_0\leq\mathcal R[R_{\mathrm{trial}}]}.
$$
Admissible trial functions must obey the outer <boundary condition>, be nonzero and have finite <weighted norm> and <energy>; they need not solve the differential equation. If the diffusion operator has no lowest <eigenfunction>, the same variational argument bounds the infimum of its <spectrum>. Existence of a <fundamental diffusion mode> is an additional assumption for strongly vanishing diffusivity, not a consequence of the identity alone.
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