This thesis aims to investigate characteristics and application of Laplacian Equation. First, based on fundamental solutions of Laplacian Equation, we can derive Green’s representation formula and get mean value formula for harmonic functions. Through exploring harmonic formulas, we can further realize properties of two operators, reduced operator and biharmonic operator, respectively. Second, we adapt and make somewhat adjustment maximum principles on bounded domain and enable maximum principles to apply in unbounded domain. Third, we take advantages of Poisson integral formula to prove two theorems, Schewa reflection principle and Harnack’s inequality. Fourth, we state Existence Theory to acquire the simple sufficient condition of solvability for Laplace equation. Last but not least, we illustrate realistic application of Laplacian Equation in Physics, such as Newtonian gravitation. With deep realization toward characteristics of Laplaician Equation, we are able to offer strict mathematical prove for Newtonian gravitation.

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[G-T] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2001.