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

Abstract

Starting from the approach to the Laplacian with respect to coupling measures and undirected weighted graphs, we provide a setting for a general point of view for a Kirchhoff type divergence and a Laplace operators built on the trivial gradient of order zero $f(y)-f(x)$. We consider some particular classical and new instances of this approach.