Consider \begin{align*} \Delta u(P) = 0, \quad P\in \Omega,\\ \frac{\partial u(P)}{\partial n_{p}} = f(P),\quad P\in \Gamma, \end{align*} where $\Omega$ is the infinite domain in the plane outside the boundary $\Gamma$.
Question:
Assume we prescribe $|u(P)|\to 0$ as $P \to \infty$, show that this is equivalent to \begin{equation} \int\limits_{\Gamma} f(Q)\, dS_{Q} = 0. \end{equation}
Attempt at solution:
Assume that $u$ and $\Gamma$ are sufficiently smooth, and consider the Laplace equation on $\Omega_{\epsilon} = \Omega \cap B_{\epsilon}(P_{o})$, with $P_{o}\in \Omega^{\mathsf{c}}$ and $\epsilon$ such that $\Omega^{\mathsf{c}} \subset B_{\epsilon}(P_{o})$. Therefore one has \begin{equation} 0 = \int\limits_{\Omega_{\epsilon}} \Delta u \, dV = \underbrace{\int\limits_{\partial B_{\epsilon}(P_{o})} \frac{\partial u(Q)}{\partial n_{Q}}\, dS_{Q}}_{I} + \int\limits_{\Gamma} f(Q) \, dS_{Q}, \end{equation} where it remains to show $I = 0$ as $\epsilon \to \infty$, due to $|u(P)|\to 0$ as $P \to \infty$. Where do I go from here?