Establishing a relationship between weak solution in $L_2(\Omega)$ and weak solution in $W^{1,\ 2}(\Omega)$ with classical solution I have a problem:

For $\Omega$ be a bounded domain in $\Bbb R^n$.
Denote $L_{2,\ 0}(\Omega)=\overline{C_{0}^{\infty}(\Omega)}$, with norm in $L_2(\Omega)$.
We consider:
  $$\left\{\begin{matrix}
\Delta u +u_x+au=f(x)\ \text{in}\ \Omega& \\ 
 \ \ \ \ \ \ \ \ \ \ \  u\mid_{\partial \Omega}=0\ \text{on}\ \Omega & 
\end{matrix}\right. \tag I$$
a/ Definition of weak solution of (I) in $L_2(\Omega)$;
b/ Establishing a relationship between weak solution in $L_2(\Omega)$ and weak solution in $W^{1,\ 2}(\Omega)$ with classical solution.

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We have:
$\Delta u +u_x+au=f
\iff \int_{\Omega}\Delta u \cdot v \rm dx +\int_{\Omega} u_x v \rm dx +\int_{\Omega}au \cdot v \rm dx =\int_{\Omega}fv \rm dx$
On the other hand, $$\begin{align}\int_{\Omega}\Delta u \cdot v \rm dx &=\sum_{i=1}^{n}\int_{\Omega}\dfrac{\partial^2 u}{\partial x_i^2}vdx\\ &=\int_{\partial \Omega}\sum_{i=1}^{n}\dfrac{\partial u}{\partial x_i}v\cos(n,x_i)dx-\int_{\Omega}\sum_{i=1}^{n}\dfrac{\partial u }{\partial x_i}\dfrac{\partial v}{\partial x_i}dx\\
&=-\int_{\Omega}\sum_{i=1}^{n}\dfrac{\partial u }{\partial x_i}\dfrac{\partial v}{\partial x_i}dx \end{align}$$
Whence, 
$$-\int_{\Omega}\sum_{i=1}^{n}\dfrac{\partial u }{\partial x_i}\dfrac{\partial v}{\partial x_i}dx+\int_{\Omega} u_x v \rm dx +\int_{\Omega}au \cdot v \rm dx =\int_{\Omega}fv \rm dx$$
It means that $$\boxed{\int_{\Omega}\nabla u\cdot \nabla v \rm dx-\int_{\Omega}u_xvdx-\int_{\Omega}au \cdot vdx=-\int_{\Omega}fvdx}$$
Now, I have stuch although I have tried...
Any help will be appreciated! Thanks!
 A: a) Let the notation $u_x$ imply some partial derivative, say, $\frac{\partial u}{\partial x_m}.$ The following three definitions are standard in the $L^p$-theory of boundary value problems for PDE.
Definition 1. For a bvp (I), a strong solution of the class $W^{2,2}$ is an element $u\in W^{2,2}(\Omega)$ satisfying the equation (I) a.e. in $\Omega$, with the boundary condition (I) understood in terms of traces.
Definition 2. For a bvp (I), a weak solution of the class $L^2$ is an element $u\in L^2(\Omega)$ satisfying the integral identity
$$
\int\limits_{\Omega}u\Delta\,v\,dx-\int\limits_{\Omega}u\frac{\partial v}{\partial x_m}\,dx+\int\limits_{\Omega}auv\,dx=\int\limits_{\Omega}fv\,dx\quad\forall\, v\in W^{2,2}(\Omega)\,\colon \;\,v|_{\partial\Omega} =0\tag{$\ast$}$$
Remark. If the boundary condition were inhomogeneous, say, $u|_{\partial\Omega}=\varphi$, the left-hand side of the identity $\,(\ast)\,$ should contain an additional term 
$$-\int\limits_{\partial\Omega}\varphi\frac{\partial v}{\partial n}\,ds $$ 
with  an outward  unit  normal $n$ to $\partial\Omega$.
Definition 3. For a bvp (I), a definition of a weak solution of the class $L^2$ 
is rated as correct iff
 i) a strong solution of the class $W^{2,2}$ will be a weak solution of the class $L^2$;
ii) a weak solution of the class $L^2$ possessing the smoothness of $W^{2,2}$  will be a strong solution.
Thus for Defnition 2, being correct just implies that the integral identity $\,(\ast)\,$ does contain the required equation and boundary condition. Checking the correctness of Definition 2 reduces to a not very difficult exercise. Besides the fact that $C_0^{\infty}(\Omega)$ is  dense in $L^2(\Omega)$, you need to know that for a domain $\Omega$ with a sufficiently smooth boundary, holds the following extension theorem
$$\forall\,\varphi_0\in W^{s-\frac{1}{2},2}(\partial\Omega),\;\forall\,\varphi_1\in W^{s-\frac{3}{2},2}(\partial\Omega)\;\exists\; \psi\in W^{s,2}(\Omega)\;\colon\; \psi|_{\partial\Omega}=\varphi_0,\;\frac{\partial\psi}{\partial n}\biggr|_{\partial\Omega}=\varphi_1\,, $$ 
where $s>\frac{3}{2}\,$,  though $\,s=2\,$ is quite enough here. You will need this this extension theorem to extract the boundary condition $\,u|_{\partial\Omega}=0\,$ from the integral identity $\,(\ast)\,$.
b) Establishing a relationship between weak solutions of the class $L^2$  and classical solutions is a rather technical matter based on the so-called localization techniques that cannot fit this format.
