A nonlocal 1-Laplacian problem and median values

• Autores: José Manuel Mazón Ruiz, Mayte Pérez Llanos, Julio D. Rossy, José Julián Toledo Melero
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 60, Nº 1, 2016, págs. 27-53
• Idioma: inglés
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• Resumen

  • In this paper, we study solutions to a nonlocal 1-Laplacian equation. 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.