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PreprintCC BY 4.0arXiv

Volumetric density estimates for nonlocal minimal surfaces

  • Mateusz Kwaśnicki
  • Jack Thompson
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Abstract

In this article, we prove that viscosity subsolutions to nonlocal mean curvature-type equations satisfy universal volumetric estimates at all scales provided their boundary has zero Lebesgue measure. Our results hold for a general class of symmetric kernels including kernels that are comparable to the fractional Laplacian. Furthermore, we prove that subsolutions with low density (with respect to a universal constant) necessarily have `fat boundary', that is, have topological boundary with positive Lebesgue measure.

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Cite

BibTeXRIS
BibTeX
@misc{kwasnicki2026volumetric,
  title = {{Volumetric density estimates for nonlocal minimal surfaces}},
  author = {Mateusz Kwaśnicki and Jack Thompson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.02176},
  eprint = {2605.02176},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.02176},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Volumetric density estimates for nonlocal minimal surfaces
AU  - Mateusz Kwaśnicki
AU  - Jack Thompson
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2605.02176
UR  - https://arxiv.org/abs/2605.02176
AB  - In this article, we prove that viscosity subsolutions to nonlocal mean curvature-type equations satisfy universal volumetric estimates at all scales provided their boundary has zero Lebesgue measure. Our results hold for a general class of symmetric kernels including kernels that are comparable to the fractional Laplacian. Furthermore, we prove that subsolutions with low density (with respect to a universal constant) necessarily have `fat boundary', that is, have topological boundary with positive Lebesgue measure.
ER  -