How do I show continuity of the mixed and weak solution to Zaremba problem?

Question

I am interested in showing continuity/boundedness of the weak solution to the following problem pde:

\begin{align*} 0 &= \mathbf{q} + \mathbf{\nabla}u && \quad x\in \Omega,\\ 0 &= \mathbf{\nabla} \cdot \mathbf{q} && \quad x\in \Omega,\\ 0 &= u && \quad x\in \partial \Omega_D,\\ g &= \mathbf{q}\cdot \mathbf{\eta} &&\quad x\in \partial\Omega_N. \end{align*}

How can I show that the norm to the weak solution to this problem is bounded by the Neumann Data? In other words, How can I show there exists a $C$ dependent only on the domain so that

$$ \| \mathbf{q}\|_{H^{\mathrm{div}}(\Omega)} \le C \| g \|_{H^{-1/2}(\partial\Omega_N)} $$

I would especially appreciate references to papers or books. If this is handled in any of the standard references (Grisvard, or Gilbarg and Trudinger) or the like, and I have missed it, could you tell me specifically where this is handled?

To the mods, I posted this question earlier today, on math.stackexchange. I have taken it down from there as I think it is more appropriate here.

Answer 1

Step 1. (Construct a lift) Let $H^1_{00}(\Omega)$ denote the subset of $H^1(\Omega)$ that vanishes on the Dirichelet boundary. Let $H^{1/2}_{00}(\partial\Omega)$ denote the set of all traces of those functions. Then $g$ actually is a bounded linear functional on that space. Use Hahn Banach to extend $g$ from a functional on $H^{1/2}_{00}(\partial\Omega)$ to a functional $G$ on $H^{1/2}(\partial\Omega)$. Then a trace theorem (Theorem 1.3.2 in Numerical Approximation of Differential Equations by Quateroni and Valli) that there is function $\mathbf{q}_{\mathrm{lift}}\in \mathbf{H}^{\mathrm{div}}(\Omega)$ such that $\|\mathbf{q}_{\mathrm{lift}}\|_{\mathbf{H}^{\mathrm{div}}(\Omega)} \le C \|G\|_{H^{-1/2}(\Omega)} = C \|g\|_{(H^{1/2}_{00}(\partial\Omega))'}$.

Step 2. (Pose variational formulation) Split $\mathbf{q}$ as $\mathbf{q} = \mathbf{q}_z + \mathbf{q}_{\mathrm{lift}}$. Let $\mathbf{H}^{\mathrm{div}}_0(\Omega)$ denote
$$\{ \mathbf{p}\in \mathbf{H}^\mathrm{div}(\Omega)\colon \forall v\in H^{1/2}_ {00}(\partial\Omega)\quad 0 =\langle \mathbf{p}\cdot \mathbf{\eta}, v\rangle_{H^{-1/2},H^{1/2}} \}.$$ Then we write the weak form of the pde as

Find $\mathbf{q}_z\in \mathbf{H}_0^{\mathrm{div}}$, $v\in L^2$ so that for all $p\in \mathbf{H}_0^{\mathrm{div}}$ we have \begin{align*} -\int_{\Omega} \mathbf{p}\cdot \mathbf{q}_{\mathrm{lift}}\, \mathrm{d}x = \int_{\Omega} \mathbf{p} \cdot \mathbf{q}_z\, \mathrm{d}x - \int_{\Omega} (\mathbf{\nabla} \cdot \mathbf{p}) u\, \mathrm{d}x \end{align*} and for all $v\in L^2$

\begin{align*} -\int_{\Omega} v (\mathbf{\nabla} \cdot \mathbf{q}_{\mathrm{lift}})\, \mathrm{d}x. = \int_{\Omega} v (\mathbf{\nabla} \cdot \mathbf{q}_z)\, \mathrm{d}x. \end{align*}

Step 3. Now, from Lax-Milgram $\mathbf{q}_z$ is bounded by $\mathbf{q}_{\mathrm{lift}}$ so we only need to use the triangle inequality and the Step 1 to get our result.