发表机构
School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明凸域上逆谱格林核的凸性，从而对满足N>2s的每个s>0建立唯一Robin中心，并进一步证明Navier多调和情形下该中心的非退化性。
AI 中文摘要
设$\Omega\subset\mathbb{R}^N$为有界凸域，$A=-\Delta_D$为正Dirichlet拉普拉斯算子。对于满足$N>2s$的每个实数$s>0$，我们证明函数$(x,y)\longmapsto K_{s,\Omega}(x,y)^{-1/(N-2s)}$（其中$K_{s,\Omega}$为$A^{-s}$的格林核）可连续地以零延拓至对角线，并在$\Omega\times\Omega$上联合凸。我们还证明相关的正则部分可实解析地跨越对角线延拓，且相应的Robin函数严格凸并在边界处发散。因此，对于满足$N>2s$的每个实数$s>0$，存在唯一的Robin中心。这尤其适用于谱分数Dirichlet拉普拉斯算子及所有整数阶Navier多调和算子。若$\Omega$属于$C^{2,\vartheta}$类（$0<\vartheta<1$），我们进一步建立二阶刚性。对于$0<s<1$，一个加权平移-曲率恒等式给出在整个$\Omega$上$D^2R_{s,\Omega}>0$。对于整数阶，我们引入一个跨越实谱阶的有限部分加倍原理。论证基于截断平方能量的两项展开以及谱恒等式$A^{-\sigma}A^{-\sigma}=A^{-2\sigma}$。由此可知，对于满足$N>2p$的每个$p\in\mathbb{N}$，唯一的Navier多调和Robin中心是非退化的；若$N>3p$，Hessian矩阵在整个域上正定。这些二阶结果所需的边界正则性与多调和阶数无关。
英文摘要
Let $Ω\subset\mathbb{R}^N$ be a bounded convex domain and let $A=-Δ_D$ be the positive Dirichlet Laplacian. For every real $s>0$ with $N>2s$, we prove that the function \[ (x,y)\longmapsto K_{s,Ω}(x,y)^{-1/(N-2s)}, \] where $K_{s,Ω}$ is the Green kernel of $A^{-s}$, extends continuously by zero to the diagonal and is jointly convex on $Ω\timesΩ$. We also prove that the associated regular part extends real analytically across the diagonal and that the corresponding Robin function is strictly convex and diverges at the boundary. Consequently, for every real $s>0$ satisfying $N>2s$, there is a unique Robin center. This applies in particular to the spectral fractional Dirichlet Laplacian and to all integer-order Navier polyharmonic operators. If $Ω$ is of class $C^{2,\vartheta}$, $0<\vartheta<1$, we further establish second-order rigidity. For $0<s<1$, a weighted translation--curvature identity yields $D^2R_{s,Ω}>0$ throughout $Ω$. For integer orders, we introduce a finite-part doubling principle across real spectral orders. The argument is based on a two-term expansion of truncated square energies together with the spectral identity $A^{-σ}A^{-σ}=A^{-2σ}$. It follows that, for every $p\in\mathbb N$ with $N>2p$, the unique Navier polyharmonic Robin center is nondegenerate; if $N>3p$, the Hessian is positive definite throughout the domain. The boundary regularity required by these second-order results is independent of the polyharmonic order.
