Suppose $u(x,y)$ is continuous in $D=\{(x,y)| 0\le x \le 1, 0\le y \le 1\}$, $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial^2 u}{\partial x \partial y}$ is absolutely integrable, prove
$$ \sup_{(x,y) \in D} |u| \le \iint_D |u| dxdy + \iint_D \left( \left| \frac{\partial u}{\partial x} \right|+\left|\frac{\partial u}{\partial y}\right| \right) dxdy + \iint_D \left|\frac{\partial^2 u}{\partial x \partial y}\right| dxdy. $$
I have tried the Taylor formular,
$$ \sup |u|=|u(\xi_x,\xi_y)+u(x_0,y_0)-u(\xi_x,\xi_y)|=\left| u(\xi_x,\xi_y)+\frac{\partial u}{\partial x}(\xi_x,\xi_y)(x_0-\xi_x)+\frac{\partial u}{\partial y}(\xi_x,\xi_y)(y_0-\xi_y)+\frac12 \frac{\partial^2 u}{\partial x \partial y}(\eta_x,\eta_y)[(x_0-\xi_x)^2+(y_0-\xi_y)^2] \right| $$ where $u(\xi_x,\xi_y)=\iint_D u dxdy$.
but there seems to be some matters of detail.
Thanks for your help.