i need some help on this question
Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with
$\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ocupying this surface is subject to :
1) A weight force F, i am given the explicit expression at each point of $\Omega$ .
2) Pressure force on $\Gamma_{1}$, i am given the explicit epression of it.
We have the following problem :
$$-\Delta u = F \quad\text{a.e in}\quad\Omega$$
$$\nabla{u}\cdot{\overrightarrow{n}_{1}} = P \quad\text{on}\quad \Gamma_{1}$$
$$\nabla{u}\cdot{\overrightarrow{n}_{2,3}}= 0\quad\text{on}\quad\Gamma_{2}\cup\Gamma_{3}$$
$$ u = 0 \quad\text{on}\quad \Gamma_{4}$$.
The weak form of this problem is Find $u\in{V}$ such that, $\forall{\phi\in{V}}$ we have :
$$\int_{\Omega}{\nabla{u}\cdot\nabla{\phi}}\cdot{dx} = \int_{\Omega}{F\phi\cdot{dx}}+\int_{\Gamma_{1}}{P\phi\cdot{d\sigma}}$$
Where $V$ is an appropriate Set .
My question is :
How can i derive a bound for the quantity $\vert{\nabla{u(x)}}\vert_{2}$ at a.e point of $\Omega$ . Note that when using a truncating test function near a point far from the boundary, we loss the contribution of the force P .
