Dimensional restrictions of potential theory and its resolution by introduction of relative newtonian potential

Abstract

Newtonian Potential is defined as a convolution of the fundamental solution of the Laplace equation and a source function. The fundamental solutions of the Laplace equations impose restrictions on the Newtonian potentials due to the dimension dependency of their form. In order to resolve the dimensional restrictions, Relative Newtonian Potential is introduced in a unified way for all dimensions. The Newtonian potentials also represent steady states of diffusion equations with the same source. In $\R$ and $\R^2$, diffusion equations with Dirac source have no equilibrium solution. We study the validity of the relative Newtonian potentials and the steady states of diffusion equations towards the relative Newtonian potentials.