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arXiv · 2004.08529

A Bourgain-Brezis-Mironescu-D\'avila theorem in Carnot groups of step two

Nicola Garofalo·Giulio Tralli
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Abstract

In this note we prove the following theorem in any Carnot group of step two $\mathbb{G}$: \[ \underset{s\nearrow 1/2}{\lim} (1 - 2s) \mathfrak P_{H,s}(E) = \frac{4}{\sqrt \pi}\ \mathfrak P_H(E). \] Here, $\mathfrak P_H(E)$ represents the horizontal perimeter of a measurable set $E\subset \mathbb{G}$, whereas the nonlocal horizontal perimeter $\mathfrak P_{H,s}(E)$ is a heat based Besov seminorm. This result represents a dimensionless sub-Riemannian counterpart of a famous characterisation of Bourgain-Brezis-Mironescu and D\'avila.

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Nicola Garofalo, Giulio Tralli. 2020-04-18. A Bourgain-Brezis-Mironescu-D\'avila theorem in Carnot groups of step two. https://arxiv.org/abs/2004.08529

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