I am reading a paper dealing with a general elliptic PDE that I need to transform from classical formulation to weak formulation: $$\left\{\begin{matrix} - \sum_{i=1}^n \sum_{j=1}^n (a_{ij} u_{x_i})_{x_j} + a_0u=f & in \ \Omega \quad \quad \quad \quad \quad \quad \boldsymbol{(^{*})}\\ \ \ \ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \ u =0 \quad \ \ \ \ & \ in \ \partial \Omega \quad \ \quad \quad \quad \quad \quad \quad \end{matrix}\right.$$
How is this equation transformed into the weak formulation?
$$ \quad \quad \quad \int_{\Omega} \sum_{i=1}^n \sum_{j=1}^n a_{ij} u_{x_i}v_{x_j} + \int_{\Omega} a_0uv=\int_{\Omega}fv \quad \forall v \in H_0^1(\Omega) \quad \quad \quad \boldsymbol{(^{**})}$$
I am aware of the fact that these are often deduced from Green's formulas, but I can't find how to do it.
I know I must to pass the derivative from the term $(a_{ij} u_{x_i}) $ on $ \boldsymbol{(^{*})}$ to $v$ by using the fact that $v=0 \ on \ \partial \Omega$, But What is the exact identity? I would also like some reference to the formulas.
The Gauss-Green theorem (the basis of the Green identities) states that if $u,v$ are sufficiently smooth on a nice domain $\Omega$ then $$\int_\Omega \frac{\partial u}{\partial x_j} v \, dx = - \int_\Omega u \frac{\partial v}{\partial x_j} \, dx + \int_{\partial \Omega} uv \nu_j dS$$ where $\nu_j$ is the $j$th component of the external normal unit vector and $dS$ is the surface area measure.
If either $u$ or $v$ happens to vanish on the boundary $\partial \Omega$ this becomes $$\int_\Omega \frac{\partial u}{\partial x_j} v \, dx = - \int_\Omega u \frac{\partial v}{\partial x_j} \, dx.$$ Using the standard approximation procedure this remains valid if $u,v \in H^1(\Omega)$ and one or the other belongs to $H^1_0(\Omega)$.
The idea is to take (*) and multiply both sides of the equation by a function $v \in H^1_0(\Omega)$. Then integrate over $\Omega$ and apply Green's identity to arrive at (**).