I'm stuck on a problem I saw in a textbook:
Suppose $\phi$ is Lipschitz, non-negative and $c$ some positive constant. Consider the Cauchy problem on a ball $B$
\begin{align} -\Delta f = c \, \phi, \quad & \text{in } B \\ f = 0, \quad & \text{on } \partial B. \end{align}
Also for $\phi$ there holds $$-\Delta \phi - c \, \phi \leq 0.$$
(Hence $-\Delta(\phi - f) \leq 0$.)
Then there holds $$\sup_{B_{1/2}} \, (\phi - f) \leq \frac 1{|B|} \int_B \phi - f \, \mathrm d x.$$
How do I arrive at the final bound?
Thanks for your help!