The Laplace Equation

Definition 1. Among the most important and ubiquitous of all partial differential equations is Laplace’s Equation: ∆u = 0, where the Laplacian operator ∆ acts on the function u : U → R (U is open in R) by taking the sum of the unmixed partial derivatives. For example: n = 1: ∆u = ∂ 2u ∂x2 = u = 0 In this simple case, the solution u = ax + b is found by integrating twice. n = 2: ∆u = ∂ 2u ∂x1 + ∂ 2u ∂x2 = 0 Here u = u(x1, x2) and the solution is already much more difficult to obtain. .. n = N : ∆u = ∂ 2u ∂x1 + ∂ 2u ∂x2 + . . . + ∂ 2u ∂xN = ∑N i=1 ∂2u ∂xi = 0 A solution to Laplace’s equation is a function u satisfying ∆u = 0 .