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Convexity of inverse spectral Green kernels and nondegeneracy of Robin centers in convex domains
Analysis of PDEs · Submitted 24 Sep 2026
abstract compiled · 2026-09-25 · no language-model review
The authors' main claim Consequently, for every real $s>0$ satisfying $N>2s$, there is a unique Robin center.
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Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain and let $A=-\Delta_D$ be the positive Dirichlet Laplacian. For every real $s>0$ with $N>2s$, we prove that the function \[ (x,y)\longmapsto K_{s,\Omega}(x,y)^{-1/(N-2s)}, \] where $K_{s,\Omega}$ is the Green kernel of $A^{-s}$, extends continuously by zero to the diagonal and is jointly convex on $\Omega\times\Omega$. 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 $\Omega$ 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 $D^2R_{s,\Omega}>0$ throughout $\Omega$. 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 $A^{-\sigma}A^{-\sigma}=A^{-2\sigma}$. 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain and let $A=-\Delta_D$ be the positive Dirichlet Laplacian. | background 1.00 | 0.10 | 0.02 | asserted 0.82 |
| s1 | For every real $s>0$ with $N>2s$, we prove that the function \[ (x,y)\longmapsto K_{s,\Omega}(x,y)^{-1/(N-2s)}, \] where $K_{s,\Omega}$ is the Green kernel of $A^{-s}$, extends continuously by zero to the diagonal and is jointly convex on $\Omega\times\Omega$. | result 0.97 | 0.96 | 0.64 | asserted 1.00 |
| s2 | 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. | result 0.99 | 0.95 | 0.16 | asserted 1.00 |
| s3 | Consequently, for every real $s>0$ satisfying $N>2s$, there is a unique Robin center. | result 0.73 | 0.90 | 0.39 | asserted 1.00 |
| s4 | This applies in particular to the spectral fractional Dirichlet Laplacian and to all integer-order Navier polyharmonic operators. | interpretation 0.63 | 0.70 | 0.09 | asserted 0.99 |
| s5 | If $\Omega$ is of class $C^{2,\vartheta}$, $0<\vartheta<1$, we further establish second-order rigidity. | contribution 0.52 | 0.92 | 0.04 | asserted 1.00 |
| s6 | For $0<s<1$, a weighted translation--curvature identity yields $D^2R_{s,\Omega}>0$ throughout $\Omega$. | result 0.54 | 0.93 | 0.07 | asserted 1.00 |
| s7 | For integer orders, we introduce a finite-part doubling principle across real spectral orders. | method 0.78 | 0.92 | 0.04 | asserted 0.98 |
| s8 | The argument is based on a two-term expansion of truncated square energies together with the spectral identity $A^{-\sigma}A^{-\sigma}=A^{-2\sigma}$. | method 1.00 | 0.83 | 0.02 | asserted 0.97 |
| s9 | 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. | result 0.99 | 0.93 | 0.23 | asserted 1.00 |
| s10 | The boundary regularity required by these second-order results is independent of the polyharmonic order. | result 0.79 | 0.84 | 0.04 | asserted 1.00 |
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Original abstract
Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain, and let $A=-\Delta_D$ be the Dirichlet Laplacian. For every $s>0$ with $N>2s$, we prove that the Green-kernel transform $(x,y)\mapsto K_{s,\Omega}(x,y)^{-1/(N-2s)}$ extends continuously by zero to the diagonal and is jointly convex on $\Omega\times\Omega$. 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 $\Omega$ 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 $\Omega$ when $0<s<1$ or $N>3s$. 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.
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