Establishing bounds of differential equation using a maximum principle

Establishing bounds of differential equation using a maximum principle

I would like to establish that the solution of $$-\epsilon
u''_\epsilon+b(x)u'_\epsilon=f(x)$$ satisfies $$||u^{(k)}_\epsilon||\leq
C(1+\epsilon^{-k/2}),$$ where $b,f\in C^4(\bar\Omega)$, $b(x)\geq \beta>0$
for all $x\in\bar\Omega$, $u_\epsilon(0)=u_0$ and $u_\epsilon(1)=u_1$.
Here I can assume that this differential equation satisfies the maximum
principle: Assume that $\psi(0)\geq 0$ and $\psi(1)\geq 0$, then
$-\epsilon \psi''_\epsilon+b(x)\psi'_\epsilon\geq0$ for all $x\in\Omega$
implies $\psi(x)\geq 0$ for all $x\in\bar\Omega$
To do this I will apply this maximum principle on the function
$$\psi^{\pm}(x)=||f||/\beta+\max\{|u_0|,|u_1|\}\pm u_\epsilon(x)$$ and get
$\psi^{\pm}(0)\geq0$ and $\psi^{\pm}(1)\geq0$. I also get $$-\epsilon
(\psi^{\pm})''_\epsilon+b(x)(\psi^{\pm})'_\epsilon=b(x)\frac{||f||}{\beta}\pm
f(x)\geq0$$ So the maximum principle implies $\psi^{\pm}(x)\geq0$. How
should I now proceed to establish my bound?