Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry
Eriksson-Bique, Sylvester; Giovannardi, Gianmarco; Korte, Riikka; Shanmugalingam, Nageswari; Speight, Gareth (2021-11-09)
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
© 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/
https://urn.fi/URN:NBN:fi-fe2022041929594
Tiivistelmä
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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