Assume that function $v$ is satisfying Poisson equation in the circle ($\omega$) with Dirichlet condition on the border ($\partial\omega$): $$-\mu \Delta v =1\,\,\,\mbox{in}\,\,\omega\,,\,\,\,\,\,\,\,\,\,\,\,v=0\,\,\,\mbox{on}\,\,\,\partial\omega\,.$$
How can we deduce by a simple maximum principle the following estimate: \begin{equation} |v|_{L^\infty (\omega )} \leq \, \frac{(diam\,\omega)^2}{2\mu}\,? \label{w,1} \end{equation}
I know that solution has the form $v=\frac{(diam\,\omega)^2-(x^2+y^2)}{4\mu}$ but I don't understand how to get that estimate with maximum principle.