A nonlocal 1-Laplacian problem and median values

Abstract

In this paper, we study solutions to a nonlocal 1-Laplacian equation given by − Z ΩJ J(x − y) uψ(y) − u(x) |uψ(y) − u(x)| dy = 0 for x ∈ Ω with u(x) = ψ(x) for x ∈ ΩJ \ Ω. We introduce two notions of solution and prove that the weaker of the two concepts is equivalent to a nonlocal median value property, where the median is determined by a measure related to J. We also show that solutions in the stronger sense are nonlocal analogues of local least gradient functions, in the sense that they minimize a nonlocal functional. In addition, we prove that solutions in the stronger sense converge to least gradient solutions when the kernel J is appropriately rescaled.