Problem
Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with a Lipschitz continuous boundary $\partial \Omega$; additionally, let $f \in L^2(\Omega)$ and $g \in H^{1/2}(\partial \Omega)$ be given. Let $u \in H^1(\Omega)$ denote the solution to the following Poisson problem, $$-\Delta u = f \text{ on } \Omega, \quad u = g \text{ on }\partial \Omega.$$ I wish to construct an upper bound on the $H^1$ norm of $u$ in terms of the domain $\Omega$, boundary data $g$ and domain data $f$.
Current state
I have been able to deduce the equality
$$\|u\|_{H^1(\Omega)}^2= \left\| u + \dfrac{1}{2}f\right\|_{L^2(\Omega)}^2- \dfrac{1}{4}\|f\|_{L^2(\Omega)}^2+ \oint\limits_{\partial \Omega}g \dfrac{\partial u}{\partial \mathbf{n}}\ dS_x,$$
However, I have been unable to bound the normal derivative of $u$. I am not famliar enough with elliptic theory to be able to derive another bound.
Questions
Is there a bound given in elliptic theory for these types of problems? If there isn't, then what conditions are required on $f$ and $g$ such that a uniform (maximum principle) bound can be found?
Here is one possibility to deal with the Dirichlet boundary condition. Let $u_g \in H^1(\Omega)$ be a function with $u_g = g$ on $\partial\Omega$ (in the trace sense). Then, the function $v := u - u_g$ belongs to $H_0^1(\Omega)$ and satisfies $$ \int_\Omega \nabla v \cdot \nabla \varphi\,\mathrm{d}x = \int_\Omega f \varphi - \nabla u_g \cdot \nabla \varphi \,\mathrm{d}x . $$ From here, you can bound the $H^1$-norm of $u$ via the $H^1$-norm of $u_g$ and via the $L^2$-norm of $f$. Moreover, the $H^1$-norm of $u_g$ can be bounded by the $H^{1/2}$-norm of $g$.