Newtonian Potential

Abstract

This chapter commences with the study of subnewtonian kernels that share many of the properties of the Newtonian kernel $1/|x-y|^{n-2}$ on $R^n \times R^n$ related to transforming a given function into another and the determination of properties of the transformed function passed on through the kernel. Poisson’s equation $\triangle u = f$ is shown to be solvable if $f$ has adequate smoothness properties. The last section introduces Schwarz’s Reflection Principle that allows the extension of function on one side of region $\Omega$ that is symmetric relative to a hyperplane to all of $\Omega$ with a vanishing normal derivative at points of the hyperplane that are in $\Omega$.