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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?

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