I have the problem like this $$ -\Delta u = f \ \ \text{on}\ \Omega \\ u = g_1 \ \ \text{on} \ \partial \Omega_1 \\ u = g_2 \ \ \text{on} \ \partial \Omega_2 $$ If we choose
$$ V_1 = \{ \nu_1 \in H^1 : \nu_1 = 0 \ \text{on}\ \partial \Omega_1 \bigcup \partial \Omega_2 \} \\ V_2 = \{ \nu_2 \in H^1 : \nu_2 = g_1 \ \text{on}\ \partial \Omega_1 \ \text{and}\ \nu_2 = g_2 \ on\ \partial \Omega_2 \} $$
If $v\ \in V_1$ and $u\ \in \ V_2$, and decompose $u$ into three regions $u = u_{ \Omega}+ u_ {\partial \Omega_1} + u_{\partial \Omega_2}$, the week formulation of the problem using this definition would be
$$ \int_{\Omega} \nabla u \cdot\nabla \nu_1 dx = \int_{\Omega} f \nu_1 dx - \int_{\Omega} \nabla u_{g_1}\cdot\nabla \nu_1 dx - \int_{\Omega} \nabla u_{g_2}\cdot\nabla \nu_1 dx $$ Would the definition of basis function, particularly on boundaries be correct, i.e. forcing to be zero on two boundaries for $V_1$, in addition using the definition of $V_2$ for $u$ from trace theorem point of view, i.e. if there is a trace operator that $\gamma_1 u = g_1$ and $\gamma_2 u = g_2$ where $\gamma$ is trace operator?