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PreprintarXiv licencearXiv

The Cp'-regularity conjecture near p=2

  • Se-Chan Lee
  • Taehun Lee
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Abstract

We prove the Cp'-regularity conjecture in every dimension when p>2 is sufficiently close to $2$. To this end, we establish improved Hölder estimates for the gradients of p-harmonic functions. These estimates also determine the first-order asymptotics of the optimal gradient Hölder exponent in every dimension. The proof combines compactness, harmonic rigidity of the limiting profiles, and a sharp uniform gap estimate for the first variation of the gradient excess.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026cpregularity,
  title = {{The $C^p'$-regularity conjecture near $p=2$}},
  author = {Se-Chan Lee and Taehun Lee},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09966},
  eprint = {2609.09966},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09966},
}
RIS
TY  - UNPB
TI  - The C^p'-regularity conjecture near p=2
AU  - Se-Chan Lee
AU  - Taehun Lee
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09966
UR  - https://arxiv.org/abs/2609.09966
AB  - We prove the C^p'-regularity conjecture in every dimension when p>2 is sufficiently close to $2$. To this end, we establish improved Hölder estimates for the gradients of p-harmonic functions. These estimates also determine the first-order asymptotics of the optimal gradient Hölder exponent in every dimension. The proof combines compactness, harmonic rigidity of the limiting profiles, and a sharp uniform gap estimate for the first variation of the gradient excess.
ER  -