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

In this paper, we study the existence of solutions of the equation (−Δ)s1u=f in a bounded open set with Lipschitz boundary $\Omega\subset \Rn$, vanishing on $\Co \Omega$, for some given s∈(0,1), and asymptotics as p→1 of solutions of (−Δ)spu=f. We obtain existence and convergence by comparing the Lns norm of f to the sharp fractional Sobolev constant, or, when f is non-negative, the weighted fractional Cheegar constant to 1 -- in this case, the results are sharp. We further prove that solutions are "flat" on sets of positive Lebesgue measure.