Let $\Omega=\mathbb{R}^2_+=\{(x,y) \in \mathbb{R}^2; y > 0\}$, $f \in L^2(\Omega)$, $\lambda \in \mathbb{R}^*_+$, $A=(a_{ij})_{1\leq i,j \leq 2}$, $a_{ij} \in \mathbb{R}$ and there exist $\alpha > 0$ such as $$\sum_{1\leq i,j \leq 2} a_{ij} \xi_i \xi_j \geq \alpha ||\xi||^2,\quad \forall \xi=(\xi_1,\xi_2)\in \mathbb{R}^2$$
Let $v \in H^1_0(\Omega)$. For $h \neq 0$, let $$D_h v = \dfrac{v(x+h,y)-v(x,y)}{h}$$ such as $\forall \varphi \in \mathcal{D}(\Omega):\iint_{\Omega} D_h v \varphi = - \iint_{\Omega} v D_{-h} \varphi$ and $\iint_{\Omega} |D_h v|^2 dx dy \leq \iint |\dfrac{\partial v}{\partial x}(x,y)|^2 dx dy$ and $D_h(\dfrac{\partial v}{\partial x}) = \dfrac{\partial}{\partial x}(D_h v)$ and $D_h(\dfrac{\partial v}{\partial y}) = \dfrac{\partial}{\partial y}(D_h v)$
The questions are:
We denote $V=H^1_0(\Omega)$. Let $v_h = D_h u$ be a solution of the problem $$\iint A \nabla v_h \nabla \varphi + \lambda \iint v_h \varphi = - \iint f D_{-h} \varphi$$
where $u \in H^1_0(\Omega)$ is the solution of the problem $$\iint A \nabla u \cdot \nabla v + \lambda \iint u v = \iint f v$$
How we can conclude that that
i) $$\left\lVert D_h(\dfrac{\partial u}{\partial x})\right\rVert_{L^2(\Omega)} \leq C ||f||_{L^2(\Omega)}$$
ii) $$\left\lVert D_h(\dfrac{\partial u}{\partial y})\right\rVert_{L^2(\Omega)} \leq C ||f||_{L^2(\Omega)}$$
I try with $\varphi=D_h v_h$ in the variational problem, bu i dons't fount the estimation.