BV functions and nonlocal functionals in metric measure spaces

2023/10/13 by Lahti, Panu, Pinamonti, Andrea, Zhou, Xiaodan · 2 citations
#26B30 #46E36 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2310.08882

Abstract

We study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that the limits of these nonlocal functionals are comparable to the variation ‖ Df‖(Ω) or the Sobolev semi-norm ∫_Ωg_f^p dμ, 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.

Cited by

• Characterisations of Sobolev spaces and constant functions over metric spaces
• Gamma-convergence of nonlocal energies for partitions

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