A Weighted Bossel-Daners Transfer Principle and a Pure-Power Robin Faber-Krahn Inequality

We prove a weighted Bossel-Daners transfer principle for the first Robin eigenvalue of the $p$-Laplacian under $w=m^{1\mathrm{/}{p}^{\mathrm{\prime }}}$, where $p^{\mathrm{\prime }}=p\mathrm{/}(p-1)$. The argument combines double-density isoperimetry with spectral admissibility for singular weights and normalized-flux monotonicity. It uses an exact $BV$ zero-extension formula, an $L^{p^{\mathrm{\prime }}}(mdx)$ selection lemma, and a nonatomic rank map that remains well defined on positive-measure level sets. For the singular pair $$m_b(x)=\mathrm{\mid }x{\mathrm{\mid }}^b,w_b(x)=\mathrm{\mid }x{\mathrm{\mid }}^{b\mathrm{/}{p}^{\mathrm{\prime }}},$$ known power-weight isoperimetry provides the geometric input. We establish spectral admissibility up to the critical exponent $p=N$ and, for the positive radial first eigenfunction $z=z(r)$ on the centered ball $B_R$, derive the integrated singular radial equation and center asymptotics and prove directly that $${\theta }_R^{\mathrm{\prime }}(r)>0(0<r<R),{\theta }_R(r)=\frac{\mathrm{\mid }{z}^{\mathrm{\prime }}(r){\mathrm{\mid }}^{p-1}}{r^{b\mathrm{/}{p}^{\mathrm{\prime }}}z(r{)}^{p-1}},$$ with ${\theta }_R(0)=0$ and ${\theta }_R(R)=\beta$. Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ be a finite union of bounded connected Lipschitz domains whose closures are pairwise disjoint, let ${\mathrm{\Omega }}_b^{\mathrm{♯}}$ be the centered ball of equal $\mathrm{\mid }x{\mathrm{\mid }}^b$-weighted volume, and let ${\lambda }_{1,\beta }^b$ denote the first eigenvalue for the displayed pair. Consequently, $${\lambda }_{1,\beta }^b({\mathrm{\Omega }}_b^{\mathrm{♯}})\le {\lambda }_{1,\beta }^b(\mathrm{\Omega })$$ whenever $N\ge 2$, $1<p\le N$, $-p<b<0$, and $\beta >0$. The range $1<p<N$ is complementary to the previously known $p\ge N$ weighted Talenti theory; at the shared endpoint $p=N$, the present proof also permits the singular contact $0\in \mathrm{\partial }\mathrm{\Omega }$.