Nonlocal gradient, the nonlocal Laplacian and maximum principles
arXiv:2607.13161
The paper investigates a nonlocal rho‑Laplacian defined via nonlocal gradient and divergence with a radial kernel, establishes its relation to known integro‑differential elliptic operators, and proves strong and weak maximum principles under very mild kernel assumptions.
Abstract
We study the nonlocal -Laplacian, defined as the composition of the nonlocal divergence and gradient operators associated with a general radial kernel : $Î_Ïu=\mbox{div}_Ï\left(D_Ïu\right)$. Our first main contribution is to establish a precise connection between this operator and the class of integro-differential elliptic operators studied by Fernández-Real and Ros-Oton (\cite{FernandezRos}), identifying explicit conditions on the kernel that guarantee membership in this class. Our second main contribution concerns maximum and comparison principles for the -Laplacian. We establish both a strong and a weak maximum principle under conditions on that are strictly weaker than those required for membership in the integro-differential class, thereby covering a genuinely broader family of operators. The results require only minimal assumptions on the kernel, and in particular do not rely on any fractional-type comparability condition.