The problem is given as follows
\begin{equation} \begin{array}{lc} \Delta u (\textbf{x}) = h(\textbf{x}), & \textbf{x} \in \Omega ,\\ u(\textbf{x}) = f(\textbf{x}), & \textbf{x} \in \Gamma_1 ,\\ \frac{\partial u}{ \partial \nu} = g(\textbf{x}), & \textbf{x} \in \Gamma_1 ,\\ \end{array} \end{equation} where $\Gamma_1$ is a subset of $\partial\Omega$, the boundary of $\Omega$.
I want to know where I can find the proof of the following inequality (in which space there is the one)?
\begin{equation} \Vert u \Vert \leq C (\Vert h \Vert + \Vert f \Vert + \Vert g \Vert), \end{equation} where $C$ is a constant depending only on $\Omega$.
In Elliptic Partial Differential Equations of Second Order, Gilbarg-Trudinger, I found the inequality for the Dirichlet one in $W^{1,2}(\Omega)$ space, but for my problem, I failed to adapt it.