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 journalArticleScientificpeer-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,dXX) 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.

Original languageEnglish
Pages (from-to)590-632
Number of pages43
JournalJournal of Differential Equations
Volume306
DOIs
Publication statusPublished - 5 Jan 2022
MoE publication typeA1 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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