Author:
Lahti Panu,Pinamonti Andrea,Zhou Xiaodan
Abstract
AbstractWe 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 total variation $$\Vert Df\Vert (\Omega )$$
‖
D
f
‖
(
Ω
)
or the Sobolev semi-norm $$\int _\Omega g_f^p\, d\mu $$
∫
Ω
g
f
p
d
μ
, 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.
Funder
Università degli Studi di Trento
Publisher
Springer Science and Business Media LLC
Reference46 articles.
1. Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford Lecture Ser. Math. Appl., vol. 25. Oxford University Press, Oxford, viii+133 pp (2004)
2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, Oxford University Press, New York (2000)
3. Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics, vol. 17. European Mathematical Society (EMS), Zürich, xii+403 pp (2011)
4. Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Optimal Control and Partial Differential Equations, pp. 439–455. IOS, Amsterdam (2001)
5. Brena, C., Pinamonti, A.: Nguyen’s approach to Sobolev spaces in metric measure spaces with unique tangents. Available at https://arxiv.org/pdf/2304.06561.pdf