A Nonlocal 1-laplacian Problem and Median Values
نویسندگان
چکیده
In this paper, we study solutions to a nonlocal 1-Laplacian equation given by − ∫ ΩJ J(x− y) uψ(y)− u(x) |uψ(y)− u(x)| dy = 0 for x ∈ Ω, with u(x) = ψ(x) for x ∈ ΩJ \ Ω. 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. 2010 Mathematics Subject Classification: 45G10, 45J05, 47H06.
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