A NONLOCAL 1-LAPLACIAN PROBLEM AND MEDIAN VALUES

JOSE M. MAZÓN; MARIA TERESA PÉREZ PÉREZ; JULIO D. ROSSI; JULIÁN TOLEDO

Publicacions Matemàtiques

Universitat Autònoma de Barcelona

Lugar: Barcelona; Año: 2016 vol. 60 p. 27 - 53

In this paper we study solutions to a nonlocal $1-$laplacianequation given by $$-\int_{\Omega_J}J(x-y)\frac{u_\psi(y)-u(x)}{|u_\psi(y)-u(x)|}dy=0\quad\hbox{for$x\in\Omega$},$$  with $u(x)=\psi(x)$ for $x\in\Omega_J\setminus\overline\Omega$. We introduce two notions ofsolutions and prove that   the weakest ofthese two concepts of solution is equivalent to verify an equation  involving the median of the function, thatis, the value of $u$ at a point $x$ is the median of $u$ in a ball centered at $x$(with a measure related $J$). We also show that solutions in thestrongest sense are the nonlocal analogous to local least gradientfunctions, in the sense that they minimize a nonlocal functional. In addition, we prove that they converge to least gradientfunctions when the kernel $J$ is appropriately rescaled.