Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

Loading...
Thumbnail Image
Access rights
openAccess
Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2022-01-05
Major/Subject
Mcode
Degree programme
Language
en
Pages
43
590-632
Series
Journal of Differential Equations, Volume 306
Abstract
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space (X,dX,μX) satisfying a 2-Poincaré inequality. Given a bounded domain Ω⊂X with μX(X∖Ω)>0, and a function f in the Besov class B2,2θ(X)∩L2(X), we study the problem of finding a function u∈B2,2θ(X) such that u=f in X∖Ω and Eθ(u,u)≤Eθ(h,h) whenever h∈B2,2θ(X) with h=f in X∖Ω. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on Ω, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extends the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.
Description
Funding Information: S.E-B. was partially supported by the National Science Foundation (U.S.) grant No. DMS-1704215 and by the Finnish Academy under Research Postdoctoral Grant No. 330048 . R.K. was supported by Academy of Finland Grant No. 308063 . N.S. was partially supported by the National Science Foundation (U.S.) grant No. # DMS-1800161 . G.S. was supported by Simons Collaboration Grant No. 576219 . G.G. was supported by Horizon 2020 # 777822 : GHAIA and by MEC-Feder grant MTM2017-84851-C2-1-P and PID2020-118180GB-I00 , Junta de Andalucía grants A-FQM-441-UGR18 and P20-00164 , and Research Unit MNat SOMM17/6109 . Funding Information: The authors are thankful to IMPAN for hosting the semester “Geometry and analysis in function and mapping theory on Euclidean and metric measure space”, where part of this research was conducted. This work was also partially supported by the grant # 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. The authors thank Yannick Sirefor helpful discussions and for sharing an early manuscript of [3] with us. Publisher Copyright: © 2021 The Authors
Keywords
Besov space, Existence and uniqueness for Dirichlet problem, Fractional Laplacian, Metric measure space, Strong maximum principle, Traces and extensions
Other note
Citation
Eriksson-Bique, S, Giovannardi, G, Korte, R, Shanmugalingam, N & Speight, G 2022, ' Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry ', Journal of Differential Equations, vol. 306, pp. 590-632 . https://doi.org/10.1016/j.jde.2021.10.029