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

Sylvester Eriksson-Bique; Gianmarco Giovannardi; Riikka Korte; Nageswari Shanmugalingam; Gareth Speight

doi:10.1016/j.jde.2021.10.029

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

Sylvester Eriksson-Bique, Gianmarco Giovannardi, Riikka Korte^*, Nageswari Shanmugalingam, Gareth Speight

^*Corresponding author for this work

Research output: Contribution to journal › Article › Scientific › peer-review

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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,d_X,μ_X) satisfying a 2-Poincaré inequality. Given a bounded domain Ω⊂X with μ_X(X∖Ω)>0, and a function f in the Besov class B_2,2^θ(X)∩L²(X), we study the problem of finding a function u∈B_2,2^θ(X) such that u=f in X∖Ω and E_θ(u,u)≤E_θ(h,h) whenever h∈B_2,2^θ(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.

Original language	English
Pages (from-to)	590-632
Number of pages	43
Journal	Journal of Differential Equations
Volume	306
DOIs	https://doi.org/10.1016/j.jde.2021.10.029
Publication status	Published - 5 Jan 2022
MoE publication type	A1 Journal article-refereed

Keywords

Besov space
Existence and uniqueness for Dirichlet problem
Fractional Laplacian
Metric measure space
Strong maximum principle
Traces and extensions

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10.1016/j.jde.2021.10.029Licence: CC BY

Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometryFinal published version, 470 KBLicence: CC BY

Cite this

@article{4e9aeec2444245339609b6be0bbeffef,

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

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-Poincar{\'e} 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 H{\"o}lder 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.",

keywords = "Besov space, Existence and uniqueness for Dirichlet problem, Fractional Laplacian, Metric measure space, Strong maximum principle, Traces and extensions",

author = "Sylvester Eriksson-Bique and Gianmarco Giovannardi and Riikka Korte and Nageswari Shanmugalingam and Gareth Speight",

note = "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 Andaluc{\'i}a 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: {\textcopyright} 2021 The Authors",

year = "2022",

month = jan,

day = "5",

doi = "10.1016/j.jde.2021.10.029",

language = "English",

volume = "306",

pages = "590--632",

journal = "Journal of Differential Equations",

issn = "0022-0396",

publisher = "Elsevier",

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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

SN - 0022-0396

VL - 306

SP - 590

EP - 632

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

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

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Parabolic flows with variational methods

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

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Parabolic flows with variational methods

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