I'm working on the following problem laplace problem with neumann boundary condition. I was educated that such problem will have a unique solution if we apply the mean value constraint, namely, if $u$ is the solution and $\Omega$ is the domain, then we need to impose $\int_{\Omega} u ds = 0$ to the problem to ensure a solution.
My question is : How do mathematician come up with such idea? Are there any other way that serves as a replacement to dirichlet boundary condition other than mean value constraint?
Thanks a lot!