AI 中文总结
该研究针对$p$-拉普拉斯算子的第一Robin特征值，证明了加权Bossel-Daners转移原理，推导了对应Faber-Krahn不等式，补充了$1<p<N$范围的相关理论。
AI 中文摘要
我们针对$p$-拉普拉斯算子在权函数$w=m^{1/p'}$（其中$p'=p/(p-1)$）下的第一Robin特征值，证明了一个加权Bossel-Daners转移原理。该论证结合了双密度等周不等式、奇异权的谱可容许性以及归一化通量单调性，使用了精确的BV零延拓公式、$L^{p'}(m\\,dx)$选择引理，以及在正测度水平集上仍保持良定义的非原子秩映射。对于奇异对$m_b(x)=|x|^b$、$w_b(x)=|x|^{b/p'}$，已知的幂加权等周不等式提供了几何输入。我们建立了直至临界指数$p=N$的谱可容许性，并针对中心球$B_R$上的正径向第一特征函数$z=z(r)$，推导了积分型奇异径向方程和中心渐近行为，直接证明了当$0<r<R$时$\theta_R'(r)>0$，其中$\theta_R(r)=\frac{|z'(r)|^{p-1}}{r^{b/p'}z(r)^{p-1}}$，且$\theta_R(0)=0$、$\theta_R(R)=\beta$。设$\Omega\subset\mathbb{R}^N$是有界连通Lipschitz区域的有限并，其闭包两两不交，$\Omega_b^\sharp$是具有相同$|x|^b$-加权体积的中心球，$\lambda_{1,\beta}^b$表示对应对的第一特征值。当$N\ge2$、$1<p\le N$、$-p<b<0$且$\beta>0$时，可得$\lambda_{1,\beta}^{b}(\Omega_b^\sharp) \le \lambda_{1,\beta}^{b}(\Omega)$。$1<p<N$的范围与先前已知的$p\ge N$加权Talenti理论互补；在公共端点$p=N$处，本证明还允许奇异接触$0\in\partial\Omega$。
英文摘要
We prove a weighted Bossel-Daners transfer principle for the first Robin eigenvalue of the $p$-Laplacian under $w=m^{1/p'}$, where $p'=p/(p-1)$. 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 $L^{p'}(m\,dx)$ selection lemma, and a nonatomic rank map that remains well defined on positive-measure level sets. For the singular pair \[ m_b(x)=|x|^b, \qquad w_b(x)=|x|^{b/p'}, \] 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 $θ_R(0)=0$ 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 $0\in\partialΩ$.
