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Convexity of inverse spectral Green kernels and nondegeneracy of Robin centers in convex domains

  • Xiuda Liang
  • Wenjie Wang
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Abstract

Let Ω⊂ℝN be a bounded convex domain, and let A=−ΔD be the Dirichlet Laplacian. For every s>0 with N>2s, we prove that the Green-kernel transform (x,y)↦ Ks,Ω(x,y)−1/(N−2s) extends continuously by zero to the diagonal and is jointly convex on Ω×Ω. We prove that the regular part extends real analytically across the diagonal and that the associated Robin function is strictly convex and diverges to +∞ at the boundary. Hence it has a unique critical point. If Ω is of class C2,ϑ, this point is nondegenerate for every s>0 with N>2s. Moreover, the Hessian of the Robin function is positive definite throughout Ω 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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Cite

BibTeXRIS
BibTeX
@misc{liang2026convexity,
  title = {{Convexity of inverse spectral Green kernels and nondegeneracy of Robin centers in convex domains}},
  author = {Xiuda Liang and Wenjie Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.29288},
  eprint = {2609.29288},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29288},
}
RIS
TY  - UNPB
TI  - Convexity of inverse spectral Green kernels and nondegeneracy of Robin centers in convex domains
AU  - Xiuda Liang
AU  - Wenjie Wang
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.29288
UR  - https://arxiv.org/abs/2609.29288
AB  - Let Ω⊂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)↦ K_s,Ω(x,y)^−1/(N−2s) extends continuously by zero to the diagonal and is jointly convex on Ω×Ω. We prove that the regular part extends real analytically across the diagonal and that the associated Robin function is strictly convex and diverges to +∞ at the boundary. Hence it has a unique critical point. If Ω is of class C^2,ϑ, this point is nondegenerate for every s>0 with N>2s. Moreover, the Hessian of the Robin function is positive definite throughout Ω 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.
ER  -