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arXiv · 2610.27973

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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Vesa Julin, Giampiero Palatucci. 2026-10-20. A counterexample to the fractional Michael-Simon inequality. https://arxiv.org/abs/2610.27973

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