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, 306, 590–632. https://doi.org/10.1016/j.jde.2021.10.029

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

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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)
Language: English
Published: Elsevier, 2022
Publish Date: 2022-06-17
Description:

# 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 $$(X, d_{X}, \mu_{x})$$ 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.

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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