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

Abstract

We prove a weighted Bossel-Daners transfer principle for the first Robin eigenvalue of the $p$-Laplacian under $w=m^{1/p&#39;}$, where $p&#39;=p/(p-1)$. 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 $L^{p&#39;}(m\,dx)$ selection lemma, and a nonatomic rank map that remains well defined on positive-measure level sets. For the singular pair \[ m_b(x)=|x|^b, \qquad w_b(x)=|x|^{b/p&#39;}, \] 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 \[ θ_R&#39;(r)>0 \quad(0<r<R), \qquad θ_R(r)=\frac{|z&#39;(r)|^{p-1}}{r^{b/p&#39;}z(r)^{p-1}}, \] with $θ_R(0)=0$ and $θ_R(R)=β$. Let $Ω\subset\mathbb{R}^N$ be a finite union of bounded connected Lipschitz domains whose closures are pairwise disjoint, let $Ω_b^\sharp$ be the centered ball of equal $|x|^b$-weighted volume, and let $λ_{1,β}^b$ denote the first eigenvalue for the displayed pair. Consequently, \[ λ_{1,β}^{b}(Ω_b^\sharp) \le λ_{1,β}^{b}(Ω) \] whenever $N\ge 2$, $1<p\le N$, $-p<b<0$, and $β>0$. 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 $0\in\partialΩ$.