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Yau-type gradient estimates for p-harmonic functions on Riemannian manifolds with boundary under Dirichlet boundary condition

  • Bingqi Liu
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Abstract

A Yau-type gradient estimate is proved for positive p-harmonic functions on complete Riemannian manifolds with compact boundary, under a Ricci lower bound on the Riemannian manifold and a mean curvature lower bound on the boundary, assuming the Dirichlet condition and a sign condition on the outward normal derivative. The result extends the estimate for harmonic functions by Kunikawa and Sakurai to the full range of p-Laplace operators and yields a Liouville theorem when the curvature hypotheses are nonnegative. The cutoff method of the linear theory does not extend when the exponent differs from two, since it requires a directional Hessian of the distance to the boundary that Laplacian comparison cannot control. That cutoff is replaced here by a radial barrier at the boundary and an intrinsic maximum principle in the interior.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026yautype,
  title = {{Yau-type gradient estimates for p-harmonic functions on Riemannian manifolds with boundary under Dirichlet boundary condition}},
  author = {Bingqi Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23471},
  eprint = {2609.23471},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23471},
}
RIS
TY  - UNPB
TI  - Yau-type gradient estimates for p-harmonic functions on Riemannian manifolds with boundary under Dirichlet boundary condition
AU  - Bingqi Liu
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23471
UR  - https://arxiv.org/abs/2609.23471
AB  - A Yau-type gradient estimate is proved for positive p-harmonic functions on complete Riemannian manifolds with compact boundary, under a Ricci lower bound on the Riemannian manifold and a mean curvature lower bound on the boundary, assuming the Dirichlet condition and a sign condition on the outward normal derivative. The result extends the estimate for harmonic functions by Kunikawa and Sakurai to the full range of p-Laplace operators and yields a Liouville theorem when the curvature hypotheses are nonnegative. The cutoff method of the linear theory does not extend when the exponent differs from two, since it requires a directional Hessian of the distance to the boundary that Laplacian comparison cannot control. That cutoff is replaced here by a radial barrier at the boundary and an intrinsic maximum principle in the interior.
ER  -