weak form of the problem in two domains Let $\Omega$   be an open, bounded domain,  and a smooth internal boundary $\Gamma$  divides $\Omega$ into two open and connected
sets, $\Omega1$ and $\Omega2$, where $\Omega1$ is strictly included in $\Omega$, 
which means that $\partial\Omega=\partial\Omega2$ and $\partial\Omega\cap\Gamma=\emptyset$. 
consider the following boundary value problem:
$$\Delta u_1=0 ~~in ~~\Omega1$$
$$\Delta u_2=0 ~~in ~~\Omega2$$
$$u_2=d ~~on ~~\partial \Omega$$
$$u_2-u_1=g_1~~ on ~~\Gamma$$
$$\frac{\partial u_2}{\partial n}-\frac{\partial u_1}{\partial n}=g_2~~ on ~~\Gamma$$.
I want to find the weak form of this problem when the test space is $H^1(\Omega).$
Is it true?
Find $u \in H^1(\Omega)$
$$-\int_{\Omega1}\nabla u.\nabla v\Omega1-\int_{\Omega2}\nabla u.\nabla vd\Omega2+\int_{\partial\Omega}\frac{\partial u_2}{\partial n}vds+\int_{\Gamma}g_2v ds=0.$$
How should I use other bounadary conditions?
 A: Your definition of a weak solution is incorrect. By definition, given any classical BVP, to be correct  a definition of a weak solution should meet the following criterion: every classical solution is a weak solution, while every weak solution is necessarily a classical solution whenever it be found classically smooth. For a BVP, the correct definition of a weak solution must not be mismatched with the correct setting of the BVP in a weak formulation. In your case, the correct definition of a weak solution looks something like this. Find $u\in L^2(\Omega)$ with $u\in H^1(\Omega_j),\;j=1,2$,  satisfying boundary and transmission conditions
$$
u|_{\partial\Omega}=d,\quad u|_{\Gamma_2}-u|_{\Gamma_1}=g_1, \quad 
(\Gamma_j\overset{\rm def}=\Gamma\cap\partial\Omega_j\,, \;j=1,2),
$$
in conjunction with the integral identity
$$
-\int\limits_{\Omega_1}\nabla u\cdot\nabla v\,dx-\int\limits_{\Omega_2}\nabla u\cdot\nabla v\,dx=\int\limits_{\Gamma}g_2v\,ds+\int\limits_{\Omega_1}f_1v\,dx
+\int\limits_{\Omega_2}f_2v\,dx\quad\forall\,v\in H^1_0(\Omega),\tag{$\ast$}
$$
where the unit normal $n$ to $\Gamma$ in your classical formulation is supposed to be outward w.r.t. $\Omega_1\,$.
Remark. If need be, you can follow the above advice by Hui Zhang,
which amounts to finding $\widetilde{u}\in H^1_0(\Omega)$ satisfying the integral identity
$$
-\int\limits_{\Omega}\nabla\widetilde{u} \cdot\nabla v\,dx=\int\limits_{\Gamma}g_2v\,ds-\int\limits_{\Omega_2}\nabla w\cdot\nabla v\,dx+\int\limits_{\Omega_1}f_1v\,dx
+\int\limits_{\Omega_2}f_2v\,dx\quad\forall\,v\in H^1_0(\Omega),
$$
where $w\in H^2(\Omega_2)$ might be any given function satisfying boundary conditions
$$
w|_{\Gamma}=g_1\,,\quad w|_{\partial\Omega}=d,\quad 
\frac{\partial w}{\partial n}\Bigr|_{\Gamma}=0.
$$
In this case, solution of your problem can be written in the form 
$$
u(x)=
\begin{cases}
\widetilde{u}(x),\quad x\in \Omega_1\,,\\
\widetilde{u}(x)+w(x),\quad x\in \Omega_2.
\end{cases}
$$
This is one of the many approaches to solving your proble in the correct weak formulation $(\ast)$.
