The Laplacian and Mean and Extreme Values

Abstract

The Laplace operator is pervasive in many important mathematical models, and fundamental results such as the mean value theorem for harmonic functions, and the maximum principle for superharmonic functions are well known. Less well known is how the Laplacian and its powers appear naturally in a series expansion of the mean value of a function on a ball or sphere. This result is proven here using Taylor's theorem and explicit values for integrals of monomials on balls and spheres. This result allows for nonstandard proofs of the mean value theorem and the maximum principle. Connections are also made with the discrete Laplacian arising from finite difference discretization.