Consider a Poisson boundary value problem on a domain $\Omega$ with non-overlapping Dirichlet and Neumann boundary segments that construct the domain boundary $\partial \Omega = \Gamma_N \cup \Gamma_D$. \begin{align} -\nabla^2 u &= f\quad\ \ \ \text{on} \ \Omega \\ u &= u_D \quad \text{on}\ \Gamma_D \\ -\partial u/\partial n &= g_N \quad \text{on}\ \Gamma_N \end{align} with the weak form
$$\int_{\Omega} \nabla u \cdot \nabla v dx - \int_{\Gamma_D} \nabla u \cdot\textbf{n} v ds = \int_{\Omega} fv dx + \int_{\Gamma_N} g_Nv ds$$
Note that the second term on the LHS vanishes if we request the test function $v$ to belong to $H_0^1(\Omega)$. Yet, (spoiler alert :) ) since my post exactly questions that I decided to keep it for the time being.
As far as I understood, the weak imposition of Dirichlet BC requires the first BC to hold only weakly, i.e., for all test functions $\mu$ in a proper space $Q$ we must have
$$\int_{\Gamma_D}(u-u_D) \mu ds = 0 \hspace{1cm} \forall \mu \in Q$$ Thus the weak form will take the form of a mixed problem:
$$\int_{\Omega} \nabla u \cdot \nabla v dx + \int_{\Gamma_D} lv ds = \int_{\Omega} fv dx + \int_{\Gamma_N} g_Nv ds$$ $$\int_{\Gamma_D}u \mu ds = \int_{\Gamma_D}u_D \mu ds$$ for trial functions $(u,l)=:(u,-\partial u/ \partial n)$ and test functions $(v, \mu)$.
My question is what are the proper spaces for $u,v, l$, and $\mu$?
I'm only sure that:
- $u$ doesn't need to be in $H_{u_D}^1(\Omega)$ (otherwise, the second equation would be trivial.)
- $v$ isn't in $H_0^1(\Omega)$ (otherwise, the second term in the LHS of the first equation would vanish, leading to a mixed problem with one test function and two trials - which might make sense, but never have seen anything like that...)