Does the space $V = \{ v \in {H^1}(0,1),{v_x}(0) = {v_x}(1) = 0,\int\limits_0^1 {v = 0}\} $ make sense?

Question

Does this space make sense? $$V = \{ v \in {H^1}(0,1),{v_x}(0) = {v_x}(1) = 0,\int\limits_0^1 {v = 0}\}$$ where $H^1$ is the usual sobolev space.

Answer 1

Not really. You often see $H^1$ defined as the space of functions $f$ such that $f,f'\in L^2$. But that $f'$ is a "weak derivative"; in fact a function in $H^1$ need not be differentiable. So it doesn't really make sense to talk about $f'(0)$ for $f\in H^1$.

Of course whether the definition "makes sense" depends on exactly what you mean by the definition. You could define $V$ to be the space of all $f\in H^1$ such that $f'(0)$ and $f'(1)$ both exist and equal $0$.

That makes sense because elements of $H^1$ are continuous, hence they do have well-defined values at every point, so you can talk about whether they're differentiable at a point. This is in contrast to for example the space $L^1$; talking about the space of $f\in L^1$ such that $f(0)=0$ really makes no sense, because there's really no such thing as $f(0)$ in that case.