Open Access
2016 A Nonlocal 1-Laplacian Problem and Median Values
José M. Mazón, Mayte Pérez-Llanos, Julio D. Rossi, Julián Toledo
Publ. Mat. 60(1): 27-53 (2016).


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.


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José M. Mazón. Mayte Pérez-Llanos. Julio D. Rossi. Julián Toledo. "A Nonlocal 1-Laplacian Problem and Median Values." Publ. Mat. 60 (1) 27 - 53, 2016.


Published: 2016
First available in Project Euclid: 22 December 2015

zbMATH: 1341.45001
MathSciNet: MR3447733

Primary: 45G10 , 45J05 , 47H06

Keywords: $1$-Laplacian , least gradient functions , median value

Rights: Copyright © 2016 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.60 • No. 1 • 2016
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