Regularity of solutions to the fractional CheegerLaplacian on domains in metric spaces of bounded geometry 

Author:  ErikssonBique, Sylvester^{1}; Giovannardi, Gianmarco^{2}; Korte, Riikka^{3}; 
Organizations: 
^{1}Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 3000, FI90014, Oulu, Finland ^{2}Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 1438123 Povo (Trento), Italy ^{3}Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI00076 Aalto, Finland
^{4}Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH 452210025, USA

Format:  article 
Version:  published version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.5 MB) 
Persistent link:  http://urn.fi/urn:nbn:fife2022041929594 
Language:  English 
Published: 
Elsevier,
2022

Publish Date:  20220617 
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 2Poincaré 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 nonlocal 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:  00220396 
ISSNE:  10902732 
ISSNL:  00220396 
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/ 