I am trying to find the weak variational form of the PDE \begin{align*} - \Delta u + u &= f(x,y), \hspace{.5cm} (x,y) \in T \\ u &= g_1(x), \hspace{.5cm} (x,y) \in T_1 \\ u &= g_2(y), \hspace{.5cm} (x,y) \in T_2 \\ \frac{\partial u}{\partial n} &= h(x,y), \hspace{.5cm} (x,y) \in T_3 \\ \end{align*} where \begin{align*} T &= \left\{(x,y) \vert \hspace{.2cm} x > 0, y > 0, x+y < 1 \right\}\\ T_1 &= \left\{(x,y)\vert \hspace{.2cm} y = 0, \hspace{.1cm} < x < 1 \right\}\\ T_2 &= \left\{(x,y) \vert \hspace{.2cm} x= 0, \hspace{.1cm} 0 < y < 1 \right\}\\ T_3 &= \left\{(x,y) \vert \hspace{.2cm} x > 0, y > 0, x+y = 1 \right\}\\ \end{align*}
I figured I would define my space of test functions as $H = \left\{ v \in H^1(T) \vert v = 0 \text{ on } T_1 \cup T_2 \right\}$. Since Lax-Milgram can only be applied if the function belongs to the same space as the test functions, I need to decompose $u$ so that $u = \phi + u_{g_1} + u_{g_2}$, where $\phi \in H$. In this case, how would I go about defining $u_{g_1}$ and $u_{g_2}$ so that I can proceed with my weak formulation? I was thinking of defining $u_{g_i}$ to be harmonic on $T$ and satisfying the boundary conditions $u_{g_i}\vert_{T_i} = g_i$, $u_{g_i}\vert_{\partial T - T_i } = 0$, but I'm not sure if this is allowed. Any advice would be much appreciated!