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PreprintarXiv licencearXiv

Optimal regularity for the arrival time equation

  • Yiqi Huang
  • Jingze Zhu
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Abstract

For a convex mean curvature flow in ℝn+1, the corresponding arrival time function solves a degenerate elliptic equation that becomes singular at extinction. We prove that it is C24,α for any 0<α<1 in the plane, with a logarithmic modulus for its twenty-fourth derivatives. We prove that this bound is sharp combining regularity obstruction in our companion paper. We obtain asymptotic expansions of the rescaled flow to any prescribed order, including derivative estimates for the remainder. Combined with the work of Sesum for n≥ 2, this also gives the sharp C2,2/n regularity for convex arrival time functions in higher dimensions.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026optimal,
  title = {{Optimal regularity for the arrival time equation}},
  author = {Yiqi Huang and Jingze Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31610},
  eprint = {2609.31610},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31610},
}
RIS
TY  - UNPB
TI  - Optimal regularity for the arrival time equation
AU  - Yiqi Huang
AU  - Jingze Zhu
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31610
UR  - https://arxiv.org/abs/2609.31610
AB  - For a convex mean curvature flow in ℝ^n+1, the corresponding arrival time function solves a degenerate elliptic equation that becomes singular at extinction. We prove that it is C^24,α for any 0<α<1 in the plane, with a logarithmic modulus for its twenty-fourth derivatives. We prove that this bound is sharp combining regularity obstruction in our companion paper. We obtain asymptotic expansions of the rescaled flow to any prescribed order, including derivative estimates for the remainder. Combined with the work of Sesum for n≥ 2, this also gives the sharp C^2,2/n regularity for convex arrival time functions in higher dimensions.
ER  -