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 $w=m^{1/p^{\prime }}$, where $p^{\prime }=p/(p-1)$. 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 $L^{p^{\prime }}(mdx)$ selection lemma, and a nonatomic rank map that remains well defined on positive-measure level sets. For the singular pair \[ m_b(x)=|x|^b, \qquad w_b(x)=|x|^{b/p'}, \] 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 \[ θ_R'(r)>0 \quad(0<r<R), \qquad θ_R(r)=\frac{|z'(r)|^{p-1}}{r^{b/p'}z(r)^{p-1}}, \] with ${\theta }_R(0)=0$ and ${\theta }_R(R)=\beta$. Let $\Omega \subset {\mathbb{R}}^N$ be a finite union of bounded connected Lipschitz domains whose closures are pairwise disjoint, let ${\Omega }_b^{♯}$ be the centered ball of equal $|x{|}^b$-weighted volume, and let ${\lambda }_{1,\beta }^b$ 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 $\beta >0$. 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 $0\in \partial \Omega$.