I am working through Partial Differential Equations by Evans and I am struggling to understand a step of his proof of theorem 1 in chapter 2. He presents the following argument:
The function $\phi$ is defined as follows:
I am confused about how he establishes both inequalities in (12). In the first inequality, where does C come from, and where does $\Vert D^2 f \Vert_{L^{\infty}}$ come from? Shouldn't it be $\vert \Delta f \vert_{L^{\infty}}$ Secondly, what happens to the $\Vert D^2 f \Vert_{L^{\infty}}$ in the second inequality, and where does the $\varepsilon^2$ come from? If someone could help fill in the details for me, I would much appreciate it.