A NONLOCAL 1-LAPLACIAN PROBLEM AND MEDIAN VALUES

作者：Mazon Jose M^*; Perez Llanos Mayte^*; Rossi Julio D^*; Toledo Julian^*

来源：Publicacions Matematiques, 2016, 60(1): 27-53.

DOI：10.5565/PUBLMAT_60116_02

• 1-Laplacian
• median value
• least gradient functions

In this paper, we study solutions to a nonlocal 1-Laplacian equation given by
-integral(Omega J) J(x - y) u(psi)(y) - u(x)/vertical bar u(psi)(y) - u(x)vertical bar dy = 0 for x is an element of Omega,
with u(x) = psi(x) for x is an element of Omega(J) \ (Omega) over bar. 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.

• 出版日期2016