Accepted Paper

Inserted: 13 oct 2023
Last Updated: 1 aug 2024

Journal: J. Geom. Anal.
Year: 2024

Abstract:

We study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. We show that the limits of these nonlocal functionals are comparable to the variation $\Vert Df\Vert(\Omega)$ or the Sobolev semi-norm $\int_\Omega g_f^p\, d\mu$, which extends Euclidean results to metric measure spaces. In contrast to the classical setting, we also give an example to show that the limits are not always equal to the corresponding total variation even for Lipschitz functions.

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