The Laplacian and Mean and Extreme Values

摘要: The Laplace operator is pervasive in many important mathematical models, and fundamental results such as the Mean Value Theorem for harmonic functions, Max- imum Principle super-harmonic functions are well-known. Less well-known how Laplacian its powers appear naturally a series expansion of mean value func- tion on ball or sphere. This result proven here using Taylor's explicit values integrals monomials balls spheres. allows non-standard proofs Maximum Principle. Connections also made with discrete arising from finite difference discretization. 1. THE SERIES EXPANSION OF MEAN VALUE. (or Laplacian) ind dimensions, odinger equation some forms Navier-Stokes equations. ( u = 0) sphere, Minimum Principles sub-harmonic bounded domains, It per- haps less that very when expanded terms respect to radius sphere). Our key result, 3, can be found, example, (2) spheres R 3 . given sometimes referred Pizzetti (cf. (1)), reference earliest known occurrence (4). proof uses Green's second identity, solution Laplacian, clever recursion. In this section we provide more straightforward d , employing formulas