Let $\omega$ be a function that satisfies the Laplace's equation
$$\nabla^2 \omega = 0$$
The values $\omega$ and $\dfrac{\partial \omega}{\partial n}$ are known in the boundary, but not in the interior.
I want to find the value of $I$
$$I = \int_{\Omega} \omega^2 \ dx \ dy$$
Question
How can I transform the integral $I$ over the plane domain $D$ into a boundary integral over $C$ ?
$$I = \int_{D} \omega^2 \ dx \ dy = \oint_{C} \square \ ds$$
Motivation
I'm solving laplace's equation by using Boundary Element Method and after find $\omega$, I want some mesures from it.
Basically BEM gives us both values of $\omega$ and $\dfrac{\partial \omega}{\partial n}$ at boundary, then having $I$ in terms of $\omega$ and $\dfrac{\partial \omega}{\partial n}$ is perfect.
I achieved computing for a generic polynomial function $g(x, \ y)$:
Find an arbitrary function $u(x, \ y)$ such
$$\nabla^2 u = g$$
I can use Green's second identity to get.
$$\int_{D} \omega \cdot g \ dx \ dy = \int_{D} \omega \nabla^2 u - u \nabla^2 \omega \ dx \ dy = \oint_{C} \omega \dfrac{\partial u}{\partial n} - u \dfrac{\partial \omega}{\partial n} \ ds$$
Unfortunatelly I don't know how I can proceed for $\omega^2$. I tried to use $g = \omega$, but finding $u$ such $\nabla^2 u = \omega$ is hard