Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry |
|
Author: | Eriksson-Bique, Sylvester1; Giovannardi, Gianmarco2; Korte, Riikka3; |
Organizations: |
1Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 3000, FI-90014, Oulu, Finland 2Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 1438123 Povo (Trento), Italy 3Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland
4Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH 45221-0025, USA
|
Format: | article |
Version: | published version |
Access: | open |
Online Access: | PDF Full Text (PDF, 0.5 MB) |
Persistent link: | http://urn.fi/urn:nbn:fi-fe2022041929594 |
Language: | English |
Published: |
Elsevier,
2022
|
Publish Date: | 2022-06-17 |
Description: |
AbstractWe study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space \((X, d_{X}, \mu_{x})\) satisfying a 2-Poincaré inequality. Given a bounded domain \(\Omega \subset X\) with \(\mu_{x}(X \setminus \Omega) > 0\), and a function \(f\) in the Besov class \(B^{\theta}_{2,2}(X) \cap L^{2}(X)\), we study the problem of finding a function \( u \in B^{\theta}_{2,2}(X)\) such that \( u = f\) in \(X \setminus \Omega\) and \(\mathcal{E}_{\theta}(u,u) \leq \mathcal{E}_{\theta}(h,h)\) whenever \( h \in B^{\theta}_{2,2}(X)\) with \(h = f\) in \(X \setminus \Omega\). 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 \(\Omega\), 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. see all
|
Series: |
Journal of differential equations |
ISSN: | 0022-0396 |
ISSN-E: | 1090-2732 |
ISSN-L: | 0022-0396 |
Volume: | 306 |
Pages: | 590 - 632 |
DOI: | 10.1016/j.jde.2021.10.029 |
OADOI: | https://oadoi.org/10.1016/j.jde.2021.10.029 |
Type of Publication: |
A1 Journal article – refereed |
Field of Science: |
111 Mathematics |
Subjects: | |
Copyright information: |
© 2021 The Authors. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). |
https://creativecommons.org/licenses/by/4.0/ |