Convexity of inverse spectral Green kernels and nondegeneracy of Robin centers in convex domains
Year: 2026
Type: preprint
Abstract: Let $Ω\subset\mathbb{R}^N$ be a bounded convex domain, and let $A=-Δ_D$ be the Dirichlet Laplacian. For every $s>0$ with $N>2s$, we prove that the Green-kernel transform $(x,y)\mapsto K_{s,Ω}(x,y)^{-1/(N-2s)}$ extends continuously by zero to the diagonal and is jointly convex on $Ω\timesΩ$. We prove that the regular part extends real analytically across the diagonal and that the associated Robin function is strictly convex and diverges to $+\infty$ at the boundary. Hence it has a unique critical point. If $Ω$ is of class $C^{2,\vartheta}$, this point is nondegenerate for every $s>0$ with $N>2s$. Moreover, the Hessian of the Robin function is positive definite throughout $Ω$ when $03s$. The arguments combine convexity inequalities for Dirichlet heat kernels, a weighted boundary identity at fractional orders, and a finite-part transfer of Hessian positivity between spectral powers. In particular, the results apply to spectral fractional Dirichlet Laplacians and Navier polyharmonic operators.
Source: arXiv (Cornell University)
Authors: Liang, Xiuda, Wang, Wenjie
Institution: Central China Normal University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214285205