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PreprintCC BY 4.0arXiv

A counterexample to the fractional Michael-Simon inequality

  • Vesa Julin
  • Giampiero Palatucci
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Abstract

We show that the fractional Michael-Simon inequality does not hold for general smooth hypersurfaces and for the range of 0 < s < (n-1)/n in the case p=1. We construct a family of smooth hypersurfaces, whose fractional mean curvature is nonnegative everywhere, which violates the inequality in the limit. The same construction yields a counterexample for every 1 \leq p < \infty for the range s < (n-1)/(np).

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Cite

BibTeXRIS
BibTeX
@misc{julin2026counterexample,
  title = {{A counterexample to the fractional Michael-Simon inequality}},
  author = {Vesa Julin and Giampiero Palatucci},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27973},
  eprint = {2610.27973},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27973},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A counterexample to the fractional Michael-Simon inequality
AU  - Vesa Julin
AU  - Giampiero Palatucci
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27973
UR  - https://arxiv.org/abs/2610.27973
AB  - We show that the fractional Michael-Simon inequality does not hold for general smooth hypersurfaces and for the range of 0 < s < (n-1)/n in the case p=1. We construct a family of smooth hypersurfaces, whose fractional mean curvature is nonnegative everywhere, which violates the inequality in the limit. The same construction yields a counterexample for every 1 \leq p < \infty for the range s < (n-1)/(np).
ER  -