Boundary value problem with Neumann boundary conditions

Question

Let us consider the classical elliptic problem $$ - u'' + u = f{\text{ in }}(0,1)$$ with boundary conditions $${u_x}(0) = {u_x}(1) = 0$$ It is well known that the previous problem admits unique solution in $H^1(0,1)$, but it seems to me that we can do better if we consider the space $$V = \left\{ {v \in {H^1}(0,1),{u_x}(0) = {u_x}(1) = 0} \right\}$$ Why we ignore the boundry condition in this case? thanks.

Answer 1

Because for $u\in H^1(0,1)$, the values $u_x(0)$ and $u_x(1)$ are not well-defined, as $u_x\in L^2(0,1)$.