A standard physics-textbook approach to derive an integral equation for the electrostatic potential is to use Green's second identity. The electrostatic potential $\Phi$ satisfies Poisson's equation, $$ \Delta \Phi = - \rho(x) $$ where $\rho(x)$ is a distribution of sources (charges). To turn the above into an integral equation, we use Green's second identity with $\Phi$ as one of the functions and the fundamental solution of the Laplacian, which I will denote here as $\Psi(\mathbf{x}-\mathbf{x_0})$ as the other. Since $\Delta \Psi = \delta_{\mathbf{x_0}}$, one of the integrals collapses to $\Phi(\mathbf{x_0})$, and so we obtain an integral equation for $\Phi$ in terms of the fundamental solution convolved with the source, and a surface integral involving values of $\Phi$ and the normal derivative of $\Phi$ on the surface. For example, in 3 dimensions: $$ \Phi(\mathbf{x}) = \int_D \mathrm{d}^3x' \; \frac{\rho(\mathbf{x'})}{4\pi |\mathbf{x}-\mathbf{x'}|} + \frac{1}{4 \pi} \oint_{\partial D} \mathrm{d}^2x'\; \left [ \frac{1}{|\mathbf{x} - \mathbf{x'}|} \frac{\partial \Phi}{\partial n'} - \Phi(\mathbf{x'}) \frac{\partial ~}{\partial n'}\frac{1}{|\mathbf{x}-\mathbf{x'}|} \right ]. $$
In fluid mechanics, there is a similar result to Green's second identity that goes by the name of Lorentz's reciprocal identity. Once again, this is used to derive an integral representation for the velocity field (see e.g. Pozrikidis 1992)
The aforementioned book by Pozrikidis also claims that this can be done for any linear elliptic PDE. What is the general procedure and general theory for doing so?