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Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains

  • Jiahuan Li
  • Shujun Shi
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Abstract

Let D⊂ℝn, n≥2, be a bounded convex domain, and let uD be the torsion function for the restricted half-Laplacian. We prove that D2uD is negative definite at every point of D. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason--Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.

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Cite

BibTeXRIS
BibTeX
@misc{li2026strict,
  title = {{Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains}},
  author = {Jiahuan Li and Shujun Shi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.19586},
  eprint = {2608.19586},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.19586},
}
RIS
TY  - UNPB
TI  - Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains
AU  - Jiahuan Li
AU  - Shujun Shi
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2608.19586
UR  - https://arxiv.org/abs/2608.19586
AB  - Let D⊂ℝ^n, n≥2, be a bounded convex domain, and let u_D be the torsion function for the restricted half-Laplacian. We prove that D^2u_D is negative definite at every point of D. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason--Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.
ER  -