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PreprintCC BY 4.0arXiv

Estimates on Fully Nonlinear Elliptic Equations with Unrectifiable Exceptional Gradient Sets

  • Tuoc Phan
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Abstract

We study fully nonlinear uniformly elliptic equations that are imposed only when the gradients of solutions lie outside a closed ball and a closed purely one-unrectifiable set that can be unbounded. We prove Harnack inequalities and Hölder regularity estimates for viscosity solutions, with constants independent of the exceptional set. The proofs use the geometry of the contact sets in the cusp and barrier arguments to exclude exceptional gradients. We also provide examples and counterexamples showing the role of pure one-unrectifiability. Materials and proofs are presented in detail to ensure the paper is fully self-contained and accessible to a broad audience.

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Cite

BibTeXRIS
BibTeX
@misc{phan2026estimates,
  title = {{Estimates on Fully Nonlinear Elliptic Equations with Unrectifiable Exceptional Gradient Sets}},
  author = {Tuoc Phan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12430},
  eprint = {2610.12430},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12430},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Estimates on Fully Nonlinear Elliptic Equations with Unrectifiable Exceptional Gradient Sets
AU  - Tuoc Phan
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12430
UR  - https://arxiv.org/abs/2610.12430
AB  - We study fully nonlinear uniformly elliptic equations that are imposed only when the gradients of solutions lie outside a closed ball and a closed purely one-unrectifiable set that can be unbounded. We prove Harnack inequalities and Hölder regularity estimates for viscosity solutions, with constants independent of the exceptional set. The proofs use the geometry of the contact sets in the cusp and barrier arguments to exclude exceptional gradients. We also provide examples and counterexamples showing the role of pure one-unrectifiability. Materials and proofs are presented in detail to ensure the paper is fully self-contained and accessible to a broad audience.
ER  -