# Establishing linearity in laplace's equation with no closed-form solution

1. Aug 16, 2010

### jdmo

I'm writing a paper, and as a motivation to the forthcoming finite element modeling, I want to state, with some sort of "proof" that Laplace's equation in a heterogeneous volume:

\del (sigma \del V) = 0

exhibits linearity.

By "linearity", I mean that if a set of initial conditions (call them I_1) lead to solution V_1, and another set of initial conditions (call them I_2) lead to solution V_2, then the solution to the equation with initial conditions I_1+I_2 is V_1+V_2.

But I cannot do so in a straightforward manner because there is no closed-form solution which relates V_1 to I_1, for example. Is it enough just to say that Laplace's equation is linear and thus obeys superposition to initial conditions? I would like to make my claim more convincing.