任意维度下Neumann-Poincaré算子的严格等周不等式
A sharp isoperimetric inequality for the Neumann--Poincaré operator in every dimension
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中文总结 AI 辅助
该研究证明了任意维度下Neumann-Poincaré算子的严格等周不等式，在三维情形证实了Miyanishi和Suzuki的1/6猜想，揭示了等号成立的充要条件为区域是球。
中文摘要 AI 辅助
设Ω⊂ℝᵈ（d≥2）是边界为C¹,α类（0<α<1）的有界连通区域。对于伴随Neumann-Poincaré算子K*_{∂Ω}，将其特征值归一化至1/2，令λⱼ⁺(Ω)表示零均值能量空间上的上极小-极大值。我们证明∑_{j=1}^{d}λⱼ⁺(Ω)≥(d-2)/2，由此可得λ₁⁺(Ω)≥(d-2)/(2d)，等号成立当且仅当Ω为球。在三维情形，这证明了Miyanishi和Suzuki的1/6猜想。证明采用均匀外加场诱导的坐标边界电荷密度，其能量Gram矩阵为理想导体极化张量M_∞。一个2d×2d Gram矩阵的正定性给出端点Hashin-Shtrikman不等式|Ω|Tr(M_∞⁻¹)≤1，并限定K*_{∂Ω}到外加场空间的压缩迹。逆迹不等式取等时内部牛顿势为二次型，牛顿定理的逆将Ω识别为椭球；谱估计取等时该势的Hessian各向同性，迫使椭球为球。
英文摘要
Let $Ω\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,α}$, where $0<α<1$. For the adjoint Neumann--Poincaré operator $K^*_{\partialΩ}$, normalised so that its distinguished eigenvalue is $1/2$, let $λ_j^+(Ω)$ denote the upper min--max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}λ_j^+(Ω)\ge \frac{d-2}{2}.$ It follows that $λ_1^+(Ω)\ge \frac{d-2}{2d},$ with equality if and only if $Ω$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin--Shtrikman inequality $|Ω|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partialΩ}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton's theorem identifies $Ω$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.
