How to choose the Testfunction-Space? Every time I want to transform a boundary-value-problem of the form $-u'' + ... = f$ to it's weak formulation, I have problems choosing the testfunction-space.
I have the feeling that in scripts the standard testfct-space is $H_0^1(\Omega)$ (of course just if $u(x)=0 \forall x \in \partial\Omega$).
Maybe it is a silly question, but why do we choose $H_0^1(\Omega)$ and not just $W_0^{1,1}$?
Is it because we want to use the great properties of Hilbert-Spaces, especially of $L^2$ and also because we have reflexiveness then, ...?
I hope some of you can give me some answers and help me understanding the theory of differential equations better!
 A: Working on a Hilbert space is nice of course, but arguably the most important property is reflexiveness, which $W_0^{1,1}$ lacks. 
Indeed, reflexive normed spaces are characterized by the property that the closed unit ball is weakly compact. This is known as Kakutani's theorem. Moreover, by the Eberlein-Smulian theorem, weak compactness and sequential weak compactness are equivalent on a Banach space, so it also follows that on a reflexive Banach space bounded sequences admit a weakly converging subsequence.
A typical approach in proving existence results for (weak) solutions to boundary value problems is to construct a functional $J$ on some function space $X$ whose minimizers are the (weak) solutions to the boundary value problem. You then consider a minimizing sequence $\left\{u_n\right\}$, i.e. such that
$$\lim_{n\to +\infty}J(u_n)=\inf_XJ $$
 and prove that it is bounded. If $X$ is reflexive, then by the above property $\left\{u_n\right\}$ has a weakly converging subsequence $\left\{u_{n_k}\right\}\to u_0\in X$. Then, if you also prove that $J$ is weakly lower semicontinuous (which is not particularly restrictive - for instance the norm is always weakly lower semicontinuous), you get that
$$\inf_X J=\liminf_{k\to +\infty}J(u_{n_k})\geq J(u_0) $$
which implies that $u_0$ is a minimizer for $J$, and hence a (weak) solution to the associated boundary value problem.
