TY - JOUR

T1 - Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

AU - Eriksson-Bique, Sylvester

AU - Giovannardi, Gianmarco

AU - Korte, Riikka

AU - Shanmugalingam, Nageswari

AU - Speight, Gareth

N1 - 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 Sire for helpful discussions and for sharing an early manuscript of [3] with us.
Publisher Copyright:
© 2021 The Authors

PY - 2022/1/5

Y1 - 2022/1/5

N2 - 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.

AB - 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.

KW - Besov space

KW - Existence and uniqueness for Dirichlet problem

KW - Fractional Laplacian

KW - Metric measure space

KW - Strong maximum principle

KW - Traces and extensions

UR - http://www.scopus.com/inward/record.url?scp=85118902419&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2021.10.029

DO - 10.1016/j.jde.2021.10.029

M3 - Article

AN - SCOPUS:85118902419

VL - 306

SP - 590

EP - 632

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -