AI 中文总结
本文研究正交不变域中谱分数阶Laplacian的Robin函数的Hessian矩阵特征分解，证明了相关特征向量性质与非退化临界点结论，并给出类Brezis-Peletier公式的简短证明以推广结果至0<s<1的情况。
AI 中文摘要
在本研究中，我们分析正交不变域中谱分数阶Laplacian对应的Robin函数$\boldsymbol{\textit{R}}(x)$的Hessian矩阵的特征分解。我们证明：若$\boldsymbol{\textit{\u03a9}}$是在正交变换$\boldsymbol{\textit{\u039f}}$作用下不变的光滑有界凸域，则对满足$\boldsymbol{\textit{\u039f}(a)=a}$的点$\boldsymbol{\textit{\u0305}a \u2208 \boldsymbol{\textit{\u03a9}}$，梯度向量$\boldsymbol{\textit{\u2207 R}(\u0305 a)}$是雅可比矩阵$\boldsymbol{\textit{D\u039f}}$对应特征值1的特征向量。此外，若$\boldsymbol{\textit{\u03a9}}$在关于超平面$\boldsymbol{\textit{\u03c0}_v = \{x \u2208 \u211d^N : x \u22c5 v = 0\}}$的反射下不变，则存在$\boldsymbol{\textit{\u03b7 > 0}}$使得$\boldsymbol{\textit{\u210b}(\u0305 t)v = \u03b7 v}$，其中$\boldsymbol{\textit{\u210b}}$表示$\boldsymbol{\textit{R}(x)}$的Hessian矩阵。因此，若$\boldsymbol{\textit{\u03a9}}$在关于线性无关集合$\boldsymbol{\{v_1,\ldots,v_N\}}$对应的超平面$\boldsymbol{\textit{\u03c0}_{v_i}}$的反射下不变，则原点是$\boldsymbol{\textit{R}(x)}$的非退化临界点。本文还提供了类Brezis-Peletier公式的简短证明，该公式适用于$\boldsymbol{\textit{0 < s < 1}}$的任意情况，从而可推导出前述所有结果。
英文摘要
In this work we analyze the eigendecomposition of the Hessian matrix of the Robin function $\mathcal{R}(x)$ for the spectral fractional Laplacian in orthogonally invariant domains. We prove that, if $Ω$ a smooth bounded convex domain invariant under the action of an orthogonal transformation $\mathcal{O}$ then, for $\overline{t}\in\{a\inΩ:\mathcal{O}(a)=a\}$, the gradient vector $\nabla\mathcal{R}(\overline{t})$ is an eigenvector of the Jacobian $D\mathcal{O}$ associated to the eigenvalue $1$. Moreover, if $Ω$ is a domain invariant under the reflection about a hyperplane $π_{v}=\{x\in\mathbb{R}^N:x\cdot v=0\}$, there exists $η>0$ such that $\mathbb{H}(\overline{t})v=ηv$ where $\mathbb{H}$ denotes the Hessian matrix of $\mathcal{R}(x)$. Consequently, if $Ω$ is invariant under the reflection about the hyperplanes $π_{v_i}$ for a linearly independent set $\{v_1,\ldots,v_N\}$, then the origin is a non degenerate critical point of $\mathcal{R}(x)$. A short proof of the Brezis-Peletier-like formulas for $\nabla \mathcal{R}$ is also provided which allows us to prove the former results for any $0<s<1$.
