Let $\Omega$ be a bounded Lipschitz domain with unit normal $n$. Let $u\in\mathcal{C}^2(\Omega)\cap\mathcal{C}^1(\partial\Omega)$ satisfying $\Delta u = 0$. We start with [[Green's Formulae]], which reads $ \int_{\Omega}(u\,\Delta v - \Delta u\, v)\,d\Omega = \int_{\partial \Omega}(u\frac{\partial v}{\partial n}-v\frac{\partial u}{\partial n})\,d\Gamma $ for a test function $v$. Let now $u$ satisfy $\Delta u = 0$ in $\Omega$. For $v$ we choose the fundamental solution $v_x(y) = g(x, y) := \frac{1}{4\pi |x - y|}$, which satisfies $-\Delta v_x(y) = \delta_x(y)$. We hence obtain $ u(x) = \int_{\partial\Omega}g(x, y)\frac{\partial u}{\partial n(y)} - u(y)\frac{\partial g(x, y)}{\partial n(y)}d\Gamma(y) $ This can be generalized to elliptic partial differential operators $ \left(Lu\right)(x) = -\sum_{i,j=1}^d\frac{\partial}{\partial x_j}\left[a_{ji}(x)\frac{\partial u}{\partial x_i}(x)\right] $ with the fundamental solution $v_x(y)$ now satisfying $Lv_x(y) = \delta_x(y)$. # Related Items [[Green's Formulae]]