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,dX,μX) 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.

Highlights

  • In this paper we are concerned with a metric measure space (X, dX, μX)

  • Given a metric space (X, dX), a non-negative Borel function g on X is an upper gradient of a map u : X → R ∪ {−∞, ∞} if |u(γ (b)) − u(γ (a))| ≤ g ds γ for every rectifiable curve γ : [a, b] → X

  • We say that g is a 2-weak upper gradient if there is a family of curves in X such that (u, g) satisfies the above inequality for each non-constant compact rectifiable curve in X that does not belong to and Mod2( ) = 0

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Summary

Newton-Sobolev spaces and related notions

In this paper we are concerned with a metric measure space (X, dX, μX). We first start with the notion of 2-modulus of a family of curves in X. Given a metric space (X, dX), a non-negative Borel function g on X is an upper gradient of a map u : X → R ∪ {−∞, ∞} if |u(γ (b)) − u(γ (a))| ≤ g ds γ for every rectifiable curve γ : [a, b] → X. The Cartesian product Z = X × R as well as the Cartesian product Z+ = X × (0, ∞), equipped with the metric dZ and the measure μa, supports a 2-Poincaré inequality with μa doubling, see [5, Remark 4] and [7] (where we use the fact that Z+ is a uniform domain, see Proposition 4.1 below). For the setting of doubling metric measure spaces supporting a (2, 2)-Poincaré inequality, a good reference is [4, Theorem 5.53]

Cheeger differential structure
Uniform domains and co-dimension Hausdorff measures
Explicit solution
Tensorization
Tensorization of Newton-Sobolev energies
Tensorization of differentiable structures
Existence and uniqueness of Cheeger-harmonic extensions
Proof of the main theorems
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