I have a region $\Omega \subset \mathbb{R}^2$ and a harmonic function $\phi$ on $\Omega$. I know that for some "charge" $Q:\partial \Omega \to \mathbb{R}$,
$$\phi(q) = \int_{\partial \Omega} Q(p)G(p,q)\,dp,$$
where $G(p,q)$ is the Green's function of $\Delta$. Is there a way to "deconvolve" $\phi$ to recover $Q$?
Heuristic Development
We examine the behavior or $\nabla \phi$ across the boundary $\Omega$. In particular, we will heuristically evaluate the limit
$$\lim_{\nu \to 0^+}\left(\nabla \phi(\vec \rho+\hat n\nu)-\nabla \phi(\vec \rho-\hat n\nu)\right)$$
where $\hat n$ is the unit normal to the boundary $\partial \Omega$ at the point $\vec \rho$. First we write
$$\begin{align} \nabla \phi(\vec \rho+\hat n\nu)-\nabla \phi(\vec \rho-\hat n\nu)&=\int_{\partial \Omega} Q(\vec \rho')\left(\nabla G(\vec \rho+\hat n\nu|\vec \rho')-\nabla G(\vec \rho-\hat n\nu|\vec \rho')\right)d\ell'\\\\ &=\int_{\partial \Omega-C_{\delta}} Q(\vec \rho')\left(\nabla G(\vec \rho+\hat n\nu|\vec \rho')-\nabla G(\vec \rho-\hat n\nu|\vec \rho')\right)d\ell'\\\\ &+\int_{C_{\delta}} Q(\vec \rho')\left(\nabla G(\vec \rho+\hat n\nu|\vec \rho')-\nabla G(\vec \rho-\hat n\nu|\vec \rho')\right)d\ell' \tag 1 \end{align}$$
where $C_{\delta}$ is a section of $\partial \Omega$ with arc length $2\delta$ and that contains $\vec \rho$ at the "midpoint." In that which follows, the development relies on heuristic arguments, which can be made rigorous, to facilitate.
With fixed $\delta$, the first integral in the right-hand side of $(1)$ goes to zero as $\nu\to 0$ by continuity of $G(\vec \rho|\vec \rho')$ for $\vec \rho\ne \vec \rho'$.
Now, we take $\delta$ small enough so that locally we can "approximate" $C_{\delta}$ by a straight line path of length $2\delta$ (i.e., locally $C_{\delta}$ has zero curvature). Furthermore, we exploit the continuity of $Q(\vec \rho')$ on $C_{\delta}$ and write $Q(\vec \rho')\approx Q(\vec \rho)$. Then the second integral of $(1)$ can be approximated by
$$\begin{align} \int_{C_{\delta}} Q(\vec \rho')\left(\nabla G(\vec \rho+\hat n\nu|\vec \rho')-\nabla G(\vec \rho-\hat n\nu|\vec \rho')\right)d\ell'&\approx Q(\vec \rho)\int_{-\delta}^{\delta}\frac{1}{2\pi}\left(\frac{2\nu}{\rho'^2+\nu^2}\right)\,d\rho'\\\\ &=Q(\vec \rho)\frac{1}{2\pi}\left. \left(2\arctan\left(\frac{\delta}{\nu}\right)\right)\right|_{-\delta}^{\delta}\\\\ &=Q(\vec \rho)\frac{2}{\pi}\arctan\left(\frac{\delta}{\nu}\right)\\\\ &\to Q(\vec \rho)\,\,\text{as}\,\,\nu\to 0^+ \end{align}$$
Therefore, we have
$$\bbox[5px,border:2px solid #C0A000]{\lim_{\nu \to 0^+}\left(\nabla \phi(\vec \rho+\hat n\nu)-\nabla \phi(\vec \rho-\hat n\nu)\right)=Q(\vec \rho)}$$