On generalized divergence and Laplace operators as a matter of division of distributions

Abstract

Motivated by the approach to the Laplacian on undirected weighted graphs, we provide a setting for a general point of view for a Kirchhoff type divergence and a Laplace operator built on the trivial gradient f(y)−f(x) of order zero. We consider some particular classical and new instances of this approach and we introduce a notion of derivative as a tool for approximation of these operators.