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

TL;DR

This work establishes sharp existence and asymptotic results for solutions to the fractional 1-Laplacian with right-hand side $f$, by comparing either the $L^{\frac{n}{s}}$ norm of $f$ to the sharp fractional Sobolev constant $2S_{n,s}$ or a weighted Cheeger constant $h_s^f(\Omega)$. The analysis connects minimizers of the $(s,p)$-energy to the $(s,1)$-energy as $p\to1$, and identifies precise regimes where the limit problem has only the trivial solution or admits nontrivial minimizers, including sharp convergence statements for the associated energies. A key contribution is the demonstration of flatness phenomena: weak solutions and minimizers are necessarily constant on sizeable sets in a nonlocal sense, reflecting intrinsic nonlocality. The paper also introduces the crucial role of the scaling parameter $s_p= n+s-\frac{n}{p}$ in the $p\to1$ limit and establishes a comprehensive framework connecting fractional Cheeger theory, level-set minimizers, and nonlocal torsion-type behavior.

Abstract

In this paper, we study the existence of solutions of the equation $(-Δ)_1^s u=f$ in a bounded open set with Lipschitz boundary $Ω\subset \Rn$, vanishing on $\Co Ω$, for some given $s\in (0,1)$, and asymptotics as $p\to 1$ of solutions of $(-Δ)_p^s u=f$. 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.

Theorems & Definitions (53)

• Theorem 2.1
• Proposition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Proposition 3.1
• proof
• Theorem 3.2