This is a mixed boundary value problem:
\begin{align*} -\nabla \cdot(\nabla u) &= f, & x &\in \Omega \\ u &= g_D, & x &\in \Gamma_D \\ \frac{\partial u}{\partial \nu} &= g_N, & x &\in \Gamma_N \end{align*}
where $\Gamma_{D}$ is a nonempty connected subset of $\partial \Omega $ and $\Gamma_{N} = \partial \Omega - \Gamma_{D}$. I was asked to derive a suitable variational formulation and prove the resulting formulation has a unique solution. Here is my formulation.
Let $a(u, v)= \int_{\Omega}\nabla u \cdot \nabla v$, $\langle l,v \rangle = \int_{\Omega}f \cdot v + \int_{\Gamma_{N}}g_{N} \cdot v$ and $\Sigma_{g}$ = { $ v \in H^{1}(\Omega) \mid v=g \ \text{on} \ \Gamma_{D}$ }, then find $u \in \Sigma_{g_{D}}$ such that $$a(u,v) = \langle l,v \rangle \text{ for all $v$ in } \Sigma_{0}$$
But how to prove it has a unique solution? The only way I have been taught is that to use Lax Milgram theorem, that is to translate it to a problem like $$\min_{u \in V} \frac{1}{2}a(u,u) - \langle l, u \rangle$$ But here $\Sigma_{g}$ and $\Sigma_{0}$ are two function space, how can I apply Lax Milgram theorem?