Eigendecomposition of the Hessian of the Robin function in orthogonally invariant domains

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 $\mathrm{\Omega }$ a smooth bounded convex domain invariant under the action of an orthogonal transformation $\mathcal{O}$ then, for $\bar{t}\in \{a\in \mathrm{\Omega }:\mathcal{O}(a)=a\}$, the gradient vector $\mathrm{\nabla }\mathcal{R}(\bar{t})$ is an eigenvector of the Jacobian $D\mathcal{O}$ associated to the eigenvalue $1$. Moreover, if $\mathrm{\Omega }$ is a domain invariant under the reflection about a hyperplane ${\pi }_v=\{x\in {\mathbb{R}}^N:x\cdot v=0\}$, there exists $\eta >0$ such that $\mathbb{H}(\bar{t})v=\eta v$ where $\mathbb{H}$ denotes the Hessian matrix of $\mathcal{R}(x)$. Consequently, if $\mathrm{\Omega }$ is invariant under the reflection about the hyperplanes ${\pi }_{v_i}$ for a linearly independent set $\{v_1,\dots ,v_N\}$, then the origin is a non degenerate critical point of $\mathcal{R}(x)$. A short proof of the Brezis-Peletier-like formulas for $\mathrm{\nabla }\mathcal{R}$ is also provided which allows us to prove the former results for any $0<s<1$.