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

In this paper, we study the existence of solutions of the equation $(-\mathrm{\Delta }{)}_1^{s}u=f$ in a bounded open set with Lipschitz boundary $\mathrm{\Omega }\subset \text{Rn}$, vanishing on $\text{Co}\mathrm{\Omega }$, for some given $s\in (0,1)$, and asymptotics as $p\to 1$ of solutions of $(-\mathrm{\Delta }{)}_p^{s}u=f$. We obtain existence and convergence by comparing the $L^{\frac{n}{s}}$ 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.