## Publicacions Matemàtiques

### A Nonlocal 1-Laplacian Problem and Median Values

#### Abstract

In this paper, we study solutions to a nonlocal $1$-Laplacian equation given by

$$-\int_{\Omega_J} J(x-y)\frac{u_\psi(y)-u(x)}{|u_\psi(y)-u(x)|}\,dy=0\quad\text{for x\in\Omega},$$

with $u(x)=\psi(x)$ for $x\in \Omega_J\setminus\overline\Omega$. 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.

#### Article information

Source
Publ. Mat. Volume 60, Number 1 (2016), 27-53.

Dates
First available in Project Euclid: 22 December 2015