So, I got the following assignment.
Let $D\subset\mathbb{R}^n$ bounded open set woth smooth boundary. Consider the Neumann boundary problem $-\Delta u =f \text{, and } \left.\frac{\partial u}{\partial \nu}\right|_{\partial D}=0.$
(a) Define the notion of weak solution.
(b) Formulate and prove an existance theorem for weak solutions.
So, what I have done so far.
We have $-\int_D\Delta u\; v=\int_D Du\; Dv-\int_{\partial D}\frac{\partial u}{\partial \nu}v$ when ever this makes sense and by enforcing the boundary conditions we get $-\int_D\Delta u\; v=\int_D Du\; Dv$. A solution to the problem satisfies $-\int_D\Delta u\; v=\int_D f\; v$, $\forall v\in L^2(D)$, so we get $-\int_D\Delta u\; v=\int_D f\; v, \forall v\in H^1 (D).$
Let $u\sim v \Leftrightarrow u-v=c\in\mathbb{R}$ and let $V$ be the space created of choosing from every equivalent class in $H^1(D)$ the function with the smallest norm. It's eaasy to show that $V$ is Hilbert.
Anyway, I would like to use LAx-Milgram on $V$. So let $B(u,v)=\int_D Du\; Dv$. $B(\cdot,\cdot)$ is clearly bilinear. $|B(u,v)|\le ||Du||_{L^2(D)}||Dv||_{L^2(D)}\le ||u||_{H^1(D)}||v||_{H^1(D)}$.
Then assume that $B(u,u)=0\;\Rightarrow\;\int(Du)^2=0\; \Rightarrow u=c$, but by definition of $V$, $u=0$. So Lax-Milgram theorem can be applied here, which means that for every functional $F(v)=\int_Df\;v$ on $V$ there is a unique element $u$ of $V$ for which we have $B(u,v)=F(v)$.
The problem I have here is that $V$ is too big, in fact I forgot about the boundary condition. Now I would like to define $V_0=\{u\in V:\left.\frac{\partial u}{\partial \nu}\right|_{\partial D}=0\}$ but I am afraid that $\left.\frac{\partial u}{\partial \nu}\right|_{\partial D}$ is not in $L^2(\partial D)$ for all elements of $V$ but rather for only for $V\cap H^2(D)$.
Also there is a hint in the assignment that says to use the fact that if $v\in H^1(D)$ and $\int_D f=0$ then $||f||_L^2(D)\le C ||Df||_{L^2(D)}$ and I cannot see how this is useful to me.