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

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 , μ X ) satisfying a 2-Poincaré inequality. Given a bounded domain ⊂ X with μ X (X \ ) > 0, and a function f in the Besov class B θ 2,2 (X) ∩ L 2 (X), we study the problem of finding a functionWe show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on , 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.

Neumann problems for $p$-harmonic functions, and induced nonlocal operators in metric measure spaces

“…with θ ∈ (0, 1). We also prove that for p = 2, the fractional operators defined here are the same as those appearing in the literature, so that our results extend (and occasionally sharpen) earlier work by other authors in the setting of Euclidean spaces [20,58], Riemannian manifolds [8], Carnot groups [33], and even in metric measure spaces [32].…”

“…There is a conformal transformation of X to a metric space Ω with transformed metric d ρ , together with a natural tranformation ν ω of the measure ν, so that (Ω, d ρ , ν ω ) is a John domain in its completion Ω and so that (Ω, d ρ , ν ω ) satisfies our hypotheses (H0), (H1) and (H2). Moreover, Z is isometric to ∂Ω, and the fractional Laplacian (−∆ 2 ) θ as constructed in Theorem 1.10 above agrees with the construction given in [32]. This theorem will be proved in Section 7, see Subsection 7.5.…”

“…In [32] an alternate construction, based on spectral theory, for a fractional Laplacian (−∆ 2 ) θ corresponding to a Cheeger differential structure on a complete doubling metric measure space (Z, d Z , ν) supporting a 2-Poincaré inequality was constructed and studied. The methods used there depended strongly on the availability of the Poincaré inequality on the metric space itself, and moreover, it is not adaptable to fractional powers of the nonlinear p-Laplacian operator (−∆ p ) θ .…”

“…The methods used there depended strongly on the availability of the Poincaré inequality on the metric space itself, and moreover, it is not adaptable to fractional powers of the nonlinear p-Laplacian operator (−∆ p ) θ . In [32], the metric space (Z, d Z , ν) is naturally seen as the boundary of the unbounded domain Z × (0, ∞), where X = Z × [0, ∞) is equipped with the ℓ 2 -product metric and the product measure. In contrast, in our present study, we consider fractional Laplacians on the boundary of a bounded domain.…”

“…Remark 1.13. The fractional Neumann boundary value problem considered in [22, (1.6)] for bounded Euclidean domains U correspond to the solutions constructed in [32] for the domain Ω = U × (0, ∞). Theorem 1.12 shows then that the problem studied in [22, (1.6)] corresponds to the problem studied in the present paper, in the special setting of p = 2 and in the Euclidean setting.…”