Eriksson-Bique, S., Gill, J. T., Lahti, P., & Shanmugalingam, N. (2021). Asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaces. Transactions of the American Mathematical Society, 374(11): 8201-8247. https://doi.org/10.1090/tran/8495
Asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaces
|Author:||Eriksson-Bique, Sylvester1; Gill, James T.2; Lahti, Panu3;|
1Research Unit of Mathematical Sciences, P.O. Box 3000, FI-90014 Oulu, Finland
2Department of Mathematics and Statistics, Saint Louis University, Ritter Hall 307,220 N. Grand Blvd., St. Louis, Missouri 63103
3Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, People’s Republic of China
4Department of Mathematical Sciences, P.O. Box 210025, University of Cincinnati,Cincinnati, Ohio 45221-0025
|Online Access:||PDF Full Text (PDF, 0.5 MB)|
|Persistent link:|| http://urn.fi/urn:nbn:fi-fe2021092447125
American Mathematical Society,
|Publish Date:|| 2021-09-24
In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality. We show that at almost every point x outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at x. We also show that, at co-dimension 1 Hausdorff measure almost every measure-theoretic boundary point of a set (Ε) of finite perimeter, there is an asymptotic limit set Ε∞ corresponding to the asymptotic expansion of Ε and that every such asymptotic limit (Ε)∞ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of Ε∞ is Ahlfors co-dimension 1 regular.
Transactions of the American Mathematical Society
|Pages:||8201 - 8247|
|Type of Publication:||
A1 Journal article – refereed
|Field of Science:||
The first author was supported by NSF grant #DMS-1704215. The third author was partially supported by the Finnish Cultural Foundation. The fourth author was partially supported by the NSF grant #DMS-1500440 (U.S.).
© Copyright 2021 American Mathematical Society. The final authenticated version is available online at https://www.ams.org/journals/tran/0000-000-00/S0002-9947-2021-08495-0/.