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A Nonlocal 1-Laplacian Problem and Median Values

José M. Mazón, Mayte Pérez-Llanos, Julio D. Rossi, and Julián Toledo

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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

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

First available in Project Euclid: 22 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45G10: Other nonlinear integral equations 45J05: Integro-ordinary differential equations [See also 34K05, 34K30, 47G20] 47H06: Accretive operators, dissipative operators, etc.

$1$-Laplacian median value least gradient functions


Mazón, José M.; Pérez-Llanos, Mayte; Rossi, Julio D.; Toledo, Julián. A Nonlocal 1-Laplacian Problem and Median Values. Publ. Mat. 60 (2016), no. 1, 27--53.

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