I came across this problem in an exam. Given a Poisson equation on a bounded region $\Omega$,
$$\begin{cases} -\Delta u=f& u\in\Omega\\ \alpha u+\beta\dfrac{\partial u}{\partial n}=g&u\in\partial \Omega \end{cases} $$
To prove that the solution is unique, I usually use the energy method (by intergrating and applying Green formula) since it is easy to transform the boundary differentiation into other forms. But in the exam, I was asked to use the maximum principle. This link provided a way to resolve Neumann's BC using Hopf's lemma, but what about Robin/Mixed BC, there exists a plain term $\alpha u$ I don't know how to handle.
Could someone please help me?