I am wanting to show existence of solutions to
$$u_t +L(u) = f \;\;\text{on}\;\; \Omega$$
with initial condition $u|_{t=0} = u_0$ and Neumann boundary condition $\nabla u\cdot \nu = 0$ on ${\partial\Omega}$.
How do I solve this problem via the Galerkin approximation without putting the BC in the Hilbert space? It's easy to derive the weak form (which of course uses the BC) but surely this is not enough to guarantee that the solution at the end satisfies the BC, since different BCs can give rise to the same weak formulation. And in books they say the Neumann condition is only satisfied if we assume $u$ is smooth enough to do the IBP and check, but this seems like cheating.
gerw also mentioned to me that specifying this BC may not make sense since $\partial\Omega$ is measure zero. I don't know what to say about that.
Thanks
Originally posted https://math.stackexchange.com/posts/357170/
Best Answer
The Neumann boundary condition is a "natural" BC. You don't need to impose it. You have to change the space $H^1_0$ for $H^1$ and you end up with the same weak formulation that you found in the Dirichlet BC case, only that you required that the test functions lie in $H^1$.
That will do it.