Solutions of the fractional 1-Laplacian: existence, asymptotics and flatness results

In this paper, we study the existence of solutions for the equation (−Δ)^s_ 1 u = f in a bounded open set with Lipschitz boundary Ω ⊂ R^n, vanishing on R^n \ Ω, given some s ∈ (0, 1). Contextually, we obtain that the sequence of solutions for (−Δ)^s_p u = f convergences to a solution of (−Δ)^s_1 u = f when p → 1. We obtain our existence and convergence results by comparing the L^(n/s) norm of f to 1/(2S_{n,s}), where S_{n,s} is the sharp fractional Sobolev constant, or, when f is non-negative, a weighted version of the fractional Cheegar constant to 1, and in this case, the results are sharp. We further prove that solutions are “flat” on sets of positive Lebesgue measure.